2.  ÃñáììéêÝò äéáöïñéêÝò åîéóþóåéò 2çò ôÜîçò.

Ðïëëïß âáóéêïß íüìïé ôçò öõóéêÞò,ðåñéãñÜöïíôáé ìáèçìáôéêÜ  áðü äéáöïñéêÝò åîéóþóåéò äåýôåñçò ôÜîçò. 
Åéäéêüôåñá,   ïé äéáöïñéêÝò åîéóþóåéò áõôïý ôïõ ôýðïõ óõíÞèùò ðåñéãñÜöïõí öáéíüìåíá ôáëáíôþóåùí .  
              XáñáêôçñéóôéêÝò åöáñìïãÝò  óôéò ïðïßåò  óõíáíôþíôáé åîéóþóåéò áõôïý ôïõ ôýðïõ , áðïôåëïýí  ôá ëåãüìåíá óõóôÞìáôá  \  êáé ôá çëåêôñéêÜ êõêëþìáôá . 
Ç ãåíéêüôåñç ìïñöÞ, ãñáììéêÞò äéáöïñéêÞò åîßóùóçò 2  çò ôÜîçò åßíáé Þ

a(t)(((d^2) y)/(dt^2))+b (t) (dy/dt)+c (t) y=g (t),  a(t)≠0 .      (1)
		
     Ç óõíÜñôçóç g(t)  ëÝãåôáé êáé óõíÜñôçóç äéÝãåñóçò (forcing function).

ÖõóéêÜ ç åîßóùóç (1) åßíáé ðÜíôá äõíáôüí íá ãñáöåß óôç ìïñöÞ

  (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y = 
            r (t),      p(t)= ((b (t))/(a (t)))          (2)
 p(t)= ((b (t))/(a (t)))    r (t)=((g (t))/(a (t))).

Óôçí ðåñßðôùóç üðïõ  ïé óõíôåëåóôÝò    p(t)=0 ,  q(t)=0   ç åîßóùóç åðéëýåôáé ìå áðëÞ ïëïêëÞñùóç, üðùò
       èá äïýìå óôï åðüìåíï áðëü ðáñÜäåéãìá.

Ð  áñÜäåéãìá 1: Âñåßôå ôç ãåíéêÞ ëýóç ôçò åîßóùóçò   
                                     (((d^2) y)/(dt^2))=4(e^(2  t))     .         
Ëýóç:  Ç ëýóç ôçò åîßóùóçò âñßóêåôáé öõóéêÜ, ìå äýï äéáäïxéêÝò ïëïêëçñþóåéò .  Ãéá íá èõìçèïýìå êáé
    ôçí åíôïëÞ ïëïêëÞñùóçò ôïõ ÌÁÔÇÅÌÁÔÉCA  (! ) , ãñÜöïõìå ôçí åíôïëÞ ìÝóù ôçò êáôÜëëçëçò ðáëÝôáò

∫4  (E^(2  
              t)) \[DifferentialD]t

ÄçëáäÞ   y'(t)=2 (e^(2 t))+(Subscript[C, 1]), üðïõ (Subscript[C, 1]) åßíáé ç ðñþôç áõèáßñåôç óôáèåñÜ ïëïêëÞñùóçò .   Må ôç äåýôåñç ïëïêëÞñùóç:

Ýxïõìå    y (t)= (e^(2 t))+(Subscript[C, 1]) t+(Subscript[C, 2]), üðïõ (Subscript[C, 2]) åßíáé ç äåýôåñç áõèáßñåôç óôáèåñÜ ïëïêëÞñùóçò .  Ãéá ôçí åðßëõóç ôçò åîßóùóçò

ìðïñïýìå íá xñçóéìïðïéÞóïõìå ôçí åíôïëÞ ôïõ MATHEMATICA   DSolve :

s = 
      DSolve[y''[t] == 4*(E^(2  t)), y[t], 
        t]

ç ïðïßá äßíåé Üìåóá ôç ëýóç ôçò åîßóùóçò  (áðëþò õðÜñxåé åíáëëáãÞ ôùí óôáèåñþí)  .  ÂÝâáéá áðü ôï ðñþôï ìáò áðëü ðáñÜäåéãìá,  ðñÝðåé íá ôïíßóïõìå üôé ç xñÞóç ôçò åíôïëÞò  DSolve äåí ìáò äßíåé ðëçñïöïñßá ãéá ôç äéáäéêáóßá Þ ìÝèïäï åðßëõóçò. Ìðïñïýìå íá óxåäéÜóïõìå ôç ãñáöéêÞ ðáñÜóôáóç ôçò ëýóçò ãéá äéÜöïñåò ôéìÝò ôùí óôáèåñþí (Subscript[C, 1])  (Subscript[C, 2]).  Äßíïõìå ðñþôá óôç ëýóç ôçí ïíïìáóßá  f.

f = y[t] /. s[[1]]

EðåéäÞ ç ëýóç ðåñéÝxåé äýï áõèáßñåôåò óôáèåñÝò  (Subscript[C, 1])  (Subscript[C, 2]), ãéá ôï óxåäéáóìü tçò ãñáöéêÞò ðáñÜóôáóçò, èá xñçóéìïðïéÞóïõìå óõíäõáóìü ôùí åíôïëþí ôïõ MATHEMATICA Plot êáé Ôable :

Plot [
      Evaluate[
Table[
          f /. {C[1] -> i, C[2] -> i}, {i, -5, 5}], {t, -2, 
          3}], 
PlotStyle -> {RGBColor[1, 0, 0], 
          RGBColor[0, 1, 0], 
RGBColor[0, 0, 1], 
          RGBColor[1, 1, 0], RGBColor[0, 1, 1]}]

ÐáñÜäåéãìá 2: Âñåßôå ôç ãåíéêÞ ëýóç ôçò åîßóùóçò   
                                     (((d^2) y)/(dt^2))+(k^2) y =0 , üðïõ  k  åßíáé ðñáãìáôéêÞ óôáèåñÜ.       
Ëýóç: Ìðïñïýìå íá åðéëýóïõìå ôçí åîßóùóç ìå ôçí áíôéêáôÜóôáóç
                                                                                v=(dy/dt) = 
            y' .                                                                                        (a)
Xñçóéìïðïéþíôáò óôç óõíÝxåéá ôïí êáíüíá ôçò áëõóßäáò,  ëáìâÜíïõìå  ôç óxÝóç 
                                                       y'' = (dv/dt) = ((dv/dy)) ((dy/dt)) = 
            v ((dv/dy)) .                                                                      (b)
H áíôéêáôÜóôáóç ôçò óxÝóçò (a) óôçí åîßóùóç,  ôç ìåôáôñÝðåé óå åîßóùóç ðñþôçò ôÜîçò .  ÄçëáäÞ  
                                  v ((dv/dy))=+(k^2) y =0                      Þ                       vdv=-(k^2)y dy  .                                      (c)
Ïëïêëçñþíïíôáò êáé ôá äýï ìÝëç ôçò (b)   ëáìâÜíïõìå ôç óxÝóç   
                                                                                ((v^2)/2) + (((k^2) (y^2))/2) = 
        M .                                                                              (d)
H óôáèåñÜ Ì > 
          0 áöïý åßíáé ôï Üèñïéóìá äýï ôåôñáãþíùí .  ÅðïìÝíùò ìðïñïýìå íá ãñÜøïõìå 2  Ì = (k^2) 
          (L^2),  üðïõ    L åßíáé ìéá Üëëç óôáèåñÜ .  
          Tþñá áðü ôéò óxÝóåéò (a) êáé (d),  Ýxoõìå ôçí åîßóùóç  
                                                         (dy/dt) = v = ±k (Sqrt[(L^2) - (y^2)])          Þ  (dy/(Sqrt[(L^2) - (y^2)])) =   ±k dt    .                         (e)      
Tþñá ïëïêëÞñùóç êáé óôá äýï ìÝëç ôçò (e) ìáò äßíåé óáí áðïôÝëåóìá ôç óxÝóç
                                                                               arcsin ((y/L)) = ±(kt + 
              N),                                                                (f)
üðïõ Í åßíáé Üëëç ìéá óôáèåñÜ ðïõ ðñïêýðôåé áðü ôçí ïëïêëÞñùóç .  ÅðïìÝíùò ç ëýóç ôçò äéáöïñéêÞò ìáò åîßóùóçò åßíáé ç
                                                                     y=L sin(±(kt + N))=±L sin (kt + N), 
ç ïðïßá ìðïñåß êáé íá ãñáöåß óôçí áðëïýóôåñç ìïñöÞ (xñçóéìïðïéþíôáò ôïí êáíüíá ôïõ áèñüéóìáôïò ãéá ôï çìßôïíï)
                                             y = A sin (kt)  + B cos (kt),  A=±L cos (N) êáé Â=±L sin(N) .

ÐáñáôÞñçóç:Áí êÜðïéïò äåß ëßãï ðéï ðñïóåêôéêÜ ôç äéáöïñéêÞ åîßóùóç  (((d^2) y)/(dt^2))+(k^2) y=0, ßóùò ìðïñåß íá õðïèÝóåé áð' åõèåßáò üôé ïé ëýóåéò åßíáé ôçò ìïñöÞò  y (t)=sin (kt)     êáé  y (t) = cos (kt), áöïý    êáé ïé äýï áõôÝò óõíáñôÞóåéò åßíáé  áíÜëïãåò ôçò 2  çò ôïõò  ðáñáãþãïõ (ìå áñíçôéêÞ óôáèåñÜ áíáëïãßáò)! Ìå áõôü ôïí ôñüðï èá åßxáìå áðïöýãåé ôïõò õðïëïãéóìïýò ôïõ ðáñáäåßãìáôïò .   Ç äéáßóèçóç ðïëëÝò öïñÝò âïçèÜåé áí óõíäõáóôåß ìå Ýíá èåìåëéùìÝíï åðéxåßñçìá! 
   Eðßóçò ðáñáôçñïýìå üôé êáé óôá äýï ðáñáäåßãìáôá   ç ëýóç åìðëÝêåé äýï áõèáßñåôåò óôáèåñÝò .  Áõôü åßíáé ìéá xáñáêôçñéóôéêÞ éäéüôçôá ôùí ëýóåùí,  ôùí ãñáììéêþí äéáöïñéêþí åîéóþóåùí äåýôåñçò ôÜîçò.

Èá åðßëýóïõìå ìéá åîßóùóç ôçò ìïñöÞò ôïõ Ðáñáäåßãìáôïò 2, ìå ôçí åíôïëÞ    DSolve  ôïõ MATHEMATICA :  ÈÝôïõìå ãéá ðáñÜäåéãìá  k=2.

k = 2

s = DSolve[z''[t] + (k^2)*z[t] == 0, z[t], t]

Ç ëýóç ðïõ ìáò äßíåé ç åíôïëÞ DSolve åßíáé ðñÜãìáôé óùóôÞ .  ÓxåäéÜæïõìå ôþñá ôç ëýóç ãéá äéÜöïñåò ôéìÝò ôùí óôáèåñþí (Subscript[C, 1]),(Subscript[C, 2]).

p = z[t] /. s[[1]]

Plot [
      Evaluate[
Table[
          p /. {C[1] -> i, C[2] -> i}, {i, -2, 2}], {t, -6, 
          6}], 
PlotStyle -> {RGBColor[1, 0, 0], 
          RGBColor[0, 1, 0], 
RGBColor[0, 0, 1], 
          RGBColor[1, 1, 0], RGBColor[0, 1, 1]}]

Ìðïñïýìå íá åðéëýïõìå ôï ðñüâëçìá,  ãéá äéÜöïñåò ôéìÝò ôçò óôáèåñÜò k, äßíïíôáò ôçò Üëëç ôéìÞ,  êáé åêôåëþíôáò îáíÜ ôéò åíôïëÝò ôïõ ÌÁÔÇÅÌÁÔÉCA.

ÃåíéêÞ ÌÝèïäïò åðßëõóçò ãéá ôç ãñáììéêÞ äéáöïñéêÞ åîßóùóç 2çò ôÜîçò.

  BHMA 1.      ÌåôáôñÝðïõìå ôçí åîßóùóç                                                                                   
                                                                                a(t)(((d^2) y)/(dt^2))+b (t) (dy/dt)+c (t) y=g (t),  a(t)≠0 ,     (1)
                                     óôçí ìïñöÞ ôçò åîßóùóçò
                                                                                      (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y=r(t).                               (2)           

      ÂÇÌÁ 2.      Âñßóêïõìå ôç ãåíéêÞ ëýóç  ôçò ïìïãåíïýò åîßóùóçò  
                                                                                      (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y = 
            0,   
                                      ç ïðïßá åßíáé ôçò ìïñöÞò   
                            
                                                                                        (Subscript[y, ã])ï (t) =  (Subscript[C, 1]) (Subscript[y, 1]) (t) + (Subscript[C, 2]) (Subscript[y, 2]) (t),                        (Subscript[C, 1]) ,(Subscript[C, 2])  óôáèåñÝò. 

  ÂÇÌÁ 3.      Âñßóêïõìå ìéá åéäéêÞ ëýóç ôçò åîßóùóçò  (2) , Ýóôù    (Subscript[y, å])(t).



BHMA 4.       H  ãåíéêÞ ëýóç ôçò  ìÞ-ïìïãåíïýò åîßóùóçò (2), äßíåôáé áðü ôç óxÝóç

                                                                                           y (t) = (Subscript[y, ã])ï (t) +   (Subscript[y, å]) (t).

Ãéá ôçí õëïðïßçóç ôçò ìåèüäïõ, åßíáé áðáñáßôçôåò äýï ðïëý âáóéêÝò áñxÝò: Ç áñxÞ ôçò õðÝñèåóçò êáé ç áñxÞ ôçò áöáßñåóçò.

H  áñxÞ ôçò õðÝñèåóçò:        Áí   (Subscript[y, 1])(t)    êáé   (Subscript[y, 2]) (t)   åßíáé ëýóåéò ôçò ïìïãåíïýò äéáöïñéêÞò åîßóùóçò
                                                                                                    (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y = 
              0,
ôüôå åßíáé êáé ç      (Subscript[C, 1]) (Subscript[y, 1]) (t) + (Subscript[C, 2]) (Subscript[y, 2]) (t), ãéá ïðïéåóäÞðïôå óôáèåñÝò (Subscript[C, 1])  (Subscript[C, 2]) .


 H  áñxÞ ôçò áöáßñåóçò:        Áí   (Subscript[y, 1])(t)    êáé   (Subscript[y, 2]) (t)   åßíáé ëýóåéò ôçò  ìÞ- ïìïãåíïýò  äéáöïñéêÞò åîßóùóçò
                                                                                                    (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y = 
            0,
ôüôå åßíáé êáé ç      (Subscript[y, 1]) (t) - (Subscript[y, 2]) (t), åßíáé ëýóç ôçò ïìïãåíïýò åîßóùóçò
                                                                                                                                                       (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y = 
          0.

ÐáñÜäåéãìá 3: Âñåßôå ôç ãåíéêÞ ëýóç ôçò åîßóùóçò   
                                                                                  (((d^2) y)/(dt^2))+4  y =4    (a)     
Ëýóç:   Ðáñáôçñïýìå üôé ìéá åéäéêÞ ëýóç ôçò åîßóùóçò åßíáé ç   (Subscript[y, å])=1.  Áí y åßíáé ìéá Üëëç ëýóç ôçò åîßóùóçò  (a), ôüôå, å öáñìüæïíôáò ôçí áñxÞ ôçò áöáßñåóçò, ç 
                                                                                                   (Subscript[y, ï])=y-  (Subscript[y, å]),                   (b)
åßíáé ëýóç ôçò ïìïãåíïýò åîßóùóçò 
                                                                                                   (((d^2) y)/(dt^2))+4  y=0.               (c)
           ÁëëÜ áðü ôï ðáñÜäåéãìá 2, ãíùñßæïõìå üôé ç ãåíéêÞ ëýóç ôçò ïìïãåíïýò åîßóùóçò  (b),  åßíáé  ç

                                                                                  (Subscript[y, ï]) (t) =     Ácos (2  t) + 
            Bsin (2  
                    t) . 
' Áñá  xñçóéìïðïéþíôáò ôçí  (b),   ç ãåíéêÞ ëýóç ôçò åîßóùóçò (a) åßíáé ç

                                                                                   y (t) = 
          1 + Ácos (2  t) + Bsin (2  t).

1.2   Èåìåëéþäåéò ëýóåéò ôçò ïìïãåíïýò åîßóùóçò 2çò ôÜîçò  .   
      Ç ïñßæïõóá     Wronski.
            
Èá óõæçôÞóïõìå ôþñá ôç äïìÞ ôïõ óõíüëïõ ôùí ëýóåùí ôçò ïìïãåíïýò äéáöïñéêÞò åîßóùóçò 2çò ôÜîçò, 
                                                                                                     
                                                                                               y'' + p (t) y' + q (t) y = 
            0  .                (3)
     
         Ïñéóìüò  1.2.1 :      Ôï æåýãïò óõíáñôÞóåùí [(Subscript[y, 1]), 
              (Subscript[y, 2])]  ëÝãåôáé èåìåëéþäåò óýíïëï ëýóåùí ôçò åîßóùóçò (3)  óôï äéÜóôçìá   (a,  
              b)   áí êáé ïé äõï óõíáñôÞóåéò éêáíïðïéïýí ôçí åîßóùóç (3) êáé êÜèå ëýóç y (t) ôçò åîßóùóçò (3) ìðïñåß íá ãñáöåß óôç ìïñöÞ 
                                                                                            
                                                                                              y (t) = (Subscript[C, 1]) (Subscript[y, 1]) (t) + (Subscript[C, 2]) (Subscript[y, 2]) (t),                   (4)

  ãéá êÜèå t ∈ (a, 
              
              b) êáé ãéá  êáôÜëëçëåò óôáèåñÝò (Subscript[C, 1]), 
        (Subscript[C, 2]) .    Ç ëýóç ðïõ äßíåôáé áðü ôç óxÝóç (4) ãéá áõèáßñåôåò  óôáèåñÝò  (Subscript[C, 1]),  
        (Subscript[C, 2]) ëÝãåôáé ãåíéêÞ ëýóç ôçò (3)


    Óôçí ðñÜîç  åßíáé ðïëëÝò öïñÝò  äýóêïëï íá äéáðéóôþóïõìå áí åíá äïóìÝíï æåýãïò óõáíñôÞóåùí áðïôåëåß Ýíá èìåëéþäåò óýíïëï ëýóåùí ôçò äéáöïñéêÞò åîßóùóçò(3) .   Óçìáíôéêü âïÞèçìá ðáñÝxåé óå áõôÞ ôçí êáôåýèõíóç ç ïñßæïõóá ôïõ Wronski.                        

  Ïñéóìüò  1.2.2 : Ç ïñßæïõóá Wronski äýï äéáöïñßóéìùí óõíáñôÞóåùí (Subscript[y, 1,])  (Subscript[y, 2]) åßíáé ç óõíÜñôçóç W ( (Subscript[y, 1]), 
              (Subscript[y, 2])) (t) ç ïðïßá    äßíåôáé áðü ôç óxÝóç

                                                                     W ( (Subscript[y, 1]), (Subscript[y, 2])) (t) =det(({{(Subscript[y, 1]), (Subscript[y, 2])}, {((Subscript[y, 1])^′), ((Subscript[y, 2])^′)}}))(t)=(Subscript[y, 1]) (t) ((Subscript[y, 2])^′) (t)-((Subscript[y, 1])^′)(t)(Subscript[y, 2])(t).  


ÓxåôéêÜ ìå ôçí ïñßæïõóá Wronski, Ýxïõìå ôï áêüëïõèï óçìáíôéêü Èåþñçìá:


   Èåþñçìá  1.2  .3: ÕðïèÝôïõìå üôé  ôï æåýãïò  [(Subscript[y, 1]),  
                      (Subscript[y, 2]) ] éêáíïðïéåß ôç äéáöïñéêÞ åîßóùóç (3), óôï äéÜóôçìá (a, 
                  b) .   
To æåýãïò áõôü åßíáé èåìåëéþäåò óýíïëï ëýóåùí ôçò (3) áí êáé ìüíï áí W ( 
                (Subscript[y, 1]), (Subscript[y, 2])) ((Subscript[t, 0])) ≠ 0, ãéá êÜðïéï  (Subscript[t, 0])  (a, b).

ÐáñÜäåéãìá 4: Âñåßôå ôçí ïñßæïõóá Wronski ãéá ôéò óõíáñôÞóåéò    (Subscript[y, 1]) (t)=sin(t)  êáé (Subscript[y, 2])(t)=log (t)
                                                                                
Ëýóç:  Èá ïñßóïõìå ôçí ïñßæïõóá Wronski óôï Mathematica .  Èá êáôáóêåõÜóïõìå ìéá åíôïëÞ õðïëïãéóìïý  xñçóéìïðïéþíôáò ôïí Ïñéóìü 1.2  .2.

Wronskian[y1_, y2_][t_] := 
        y1[t] y2'[t] - y2[t] y1'[t]

' Exoíôáò ïñßóåé ôçí ïñßæïõóá Wronski ùò åíôïëÞ, ôçí åêôåëïýìå ãéá ôéò (Subscript[y, 1])(t)=sint    êáé  (Subscript[y, 2]) (t) = log  t, ùò åîÞò:

Wronskian[Sin, Log][t]

Mðïñïýìå ìÜëéóôá,  íá óxåäéÜóïõìå ôç ãñáöéêÞ ðáñÜóôáóç ôçò W (sint, 
            logt) :  (óôï (0.01, 
              10  ð))

Plot[{Wronskian[Sin, Log][t]}, {t, 0.01, 10  Pi}, 
      PlotStyle -> {RGBColor[1, 0, 0]}]

ÐáñÜäåéãìá 5:     Äåßîôå üôé ïé óõíáñôÞóåéò      (Subscript[y, 1]) (t)=sin5t  êáé (Subscript[y, 2]) (t) = 
            cos5t, óxçìáôßæïõí èåìåëéþäåò óýíïëï ëýóåùí ãéá ôçí åîßóùóç
                                                                                (((d^2) y)/(dt^2))+y =0 
Ëýóç:   Åßíáé åýêïëï íá äïýìå üôé ïé óõíáñôÞóåéò åßíáé ëýóåéò ôçò åîßóùóçò .  (èõìçèåßôå ôçíðáñáôÞñçóç ôïõ ðáñáäåßãìáôïò 2) .

Ç ïñßæïõóá Wronski åßíáé

Cos[2  t] D[Sin[2  t], t] - Sin[2  t] D[Cos[2  t], t]

Simplify[2 Cos[2 t]^2 + 2 Sin[2 t]^2]

ç ïðïßá åßíáé ìç - ìçäåíéêÞ ãéá êáèå t .  ÅðïìÝíùò ðñÜãìáôé ïé óõíáñôÞóåéò óxçìáôßæïõí Ýíá èåìåëéþäåò óýíïëï ëýóåùí .  ÅðïìÝíùò,  ç ãåíéêÞ ëýóç ôçò åîßóùóçò  åßíáé       y (t) = (Subscript[C, 1]) cos (2  t) + (Subscript[C, 2]) 
            sin (2  t).

1.3   EöáñìïãÞ ôçò ïñßæïõóáò Wronski :  
          H ìÝèïäïò õðïâéâáóìïý ôçò ôÜîçò
       Ìéá óçìáíôéêÞ åöáñìïãÞ ôçò ïñßæïõóáò Wronski,           óõíáíôÜôáé óôç ëåãüìåíç ìÝèïäï õðïâéâáóìïý ôçò ôÜîçò (Jean le Rond D' Alembert,  
                    1717 - 1783) .  Ç ìÝèïäïò áõôÞ åðéôñÝðåé ôçí êáôáóêåõÞ ìéáò äåýôåñçò ãñáììéêÜ áíåîÜñôçôçò ëýóçò   (Subscript[y, 2])(t) ãéá ôçí åîßóùóç  y''+p (t) y'+q (t) y=0 ,   äåäïìÝíçò ìéáò ìÞ ìçäåíéêÞò 
       ëýóçò  (Subscript[y, 1])(t). 

Ïñéóìüò  1. 3. 1 :Äýï óõíáñôÞóåéò (Subscript[y, 1])(t) êáé  (Subscript[y, 2])(t) ëÝãïíôáé ãñáììéêÜ áíåîÜñôçôåò, óå Ýíá äéÜóôçìá (a, b) áí õðÜñxïõí ìç-ìçäåíéêÝò óôáèåñÝò (Subscript[C, 1]),(Subscript[C, 2]) Ýôóé þóôå (Subscript[C, 1]) (Subscript[y, 1]) (t) + (Subscript[C, 2]) (Subscript[y, 2]) (t) = 0.  Áí äåí õðÜñxoõí ôÝôïéåò óôáèåñÝò, ïé óõíáñôÞóåéò ëÝãïíôáé ãñáììéêÜ  áíåîÜñôçôåò

Eßíáé ëïãéêü íá õðïøéáóôïýìå üôé ïé Ýííïéåò ôçò ãñáììéêÞò áíåîáñôçóßáò,  êáé ôïõ èåìåëéþäïõò óõíüëïõ ëýóåùí ãéá ôçí åîßóùóç (3),  ðñÝðåé íá óõóxåôßæïíôáé ìå ôçí ïñßæïõóá Wronski .  ÐñÜãìáôé ï óõóxåôéóìüò áõôüò,  åðéâåâáéþíåôáé áðü ôï áêüëïõèï èåþñçìá.

Èåþñçìá  1. 3. 2 : ÕðïèÝôïõìå üôé ïé óõíáñôÞóåéò (Subscript[y, 1])(t) êáé  (Subscript[y, 2])(t) åßíáé äýï öïñÝò ðáñáãùãßóéìåò óôï äéÜóôçìá (a, b) êáé üôé êáé ïé äýï åßíáé ëýóåéò ôçò äéáöïñéêÞò åîßóùóçò (3) .  Ôüôå ôá áêüëïõèá åßíáé éóïäýíáìá: 
         (i)       W ( (Subscript[y, 1]), 
                      (Subscript[y, 2])) (t)  ≠ 0, ãéá êÜèå t ∈ (a, b)
               (ii)      W( (Subscript[y, 1]), (Subscript[y, 2]))((Subscript[t, 0]))≠0, ãéá êÜðïéï (Subscript[t, 0]) ∈ (a, b)
               (iii)     Ïé óõíáñôÞóåéò (Subscript[y, 1])(t) êáé  (Subscript[y, 2])(t) åßíáé ãñáììéêÜ áíåîÜñôçôåò óôï (a, b)
               (iv)      Ïé óõíáñôÞóåéò (Subscript[y, 1])(t) êáé  (Subscript[y, 2])(t) óxçìáôßæïõí  Ýíá èåìåëéþäåò óýíïëï ëýóåùí ãéá ôçí (3).

Ãéá ôçí åöáñìïãÞ ôçò ìåèüäïõ õðïâéâáóìïý ôçò ôÜîçò,  áðáñáßôçôïò åßíáé ü ôýðïò ôïõ Abel 
     
   Èåþñçìá  1. 3. 3 : ÕðïèÝôïõìå üôé ïé óõíáñôÞóåéò  (Subscript[y, 1])(t) êáé  (Subscript[y, 2])(t) åßíáé ëýóåéò ôçò åîßóùóçò (3), óôï äéÜóôçìá (a, 
                b) .  Ôüôå ç ïñßæïõóá Wronski, ìðïñåß íá õðïëïãéóôåß áðü ôïí ôýðï ôïõ Abel       
                                               W ( (Subscript[y, 1]), (Subscript[y, 2])) (t)= W( (Subscript[y, 1]), (Subscript[y, 2]))((Subscript[t, 0]))exp(-(∫_((Subscript[t, 0]))^t) p (s) ds).             (5)

ÐáñáôÞñçóç:Áðü ôïí ôýðï (5), åßíáé ðñïöáíÞò ç éóïäõíáìßá ôùí (i) êáé  (ii) ôïõ èåùñÞìáôïò 1.3  .2.

ÌÝèïäïò Õðïâéâáóìïý ôçò ÔÜîçò ôïõ D' Alembert

                  ÕðïèÝôïõìå üôé ç óõíÜñôçóç  (Subscript[y, 1])(t) , åßíáé ìéá ìÞ - ìçäåíéêÞ ëýóç ãéá ôçí åîßóùóç (3),    óôï äéÜóôçìá (a, 
                b) .  
              Méá äåýôåñç ëýóç äßíåôáé áðü ôç óxÝóç    
                                                                       (Subscript[y, 2]) (t) =∫(( W ( (Subscript[y, 1]), (Subscript[y, 2])) (t))/((Subscript[y, 1])(((t))^2)))dt,   
  üðïõ                                                      W( (Subscript[y, 1]), (Subscript[y, 2]))(t)= C exp (-(∫_((Subscript[t, 0]))^t) p (s) 
                      ds), ãéá ìéá ïðïéáäÞðïôå ìÞ - ìçäåíéêÞ óôáèåñÜ C.

ÐáñáôÞñçóç: Ç ïíïìáóßá ôçò ìåèüäïõ  Ýxåé ïõóéáóôéêÞ óçìáóßá! : ÄåäïìÝíçò ôçò ìÞ-ìçäåíéêÞò ëýóçò  (Subscript[y, 1])(t), êáé ìå xñÞóç ôïõ ïñéóìïý ôçò ïñßæïõóáò Wronski, êáé ôïõ èåùñÞìáôïò 1.3  .2, ðñïêýðôåé  üôé ç äåýôåñç ëýóç   (Subscript[y, 2])(t), ðñÝðåé íá éêáíïðïéåß ôçí åîßóùóç 
                                                                                       ((Subscript[y, 2])^′) (t)-(((((Subscript[y, 1])^′) (t))/((Subscript[y, 1]) (t)))) (Subscript[y, 2])(t)=(W/((Subscript[y, 1])(t))),
äçëáäÞ ìéáò åîßóùóçò ðñþôçò ôÜîçò.

ÐáñÜäåéãìá 6: Âñåßôå ôç äåýôåñç ëýóç ôçò åîßóùóçò
                                                                                                   t (((d^2) y)/(dt^2))-(dy/dt)+4(t^3)y=0,
áí äßíåôáé ç ëýóç ôçò  (Subscript[y, 1]) (t)=sin((t^2)).
Ëýóç:  ÖÝñíïõìå ðñþôá ôçí åîßóùóç óôç ìïñöÞ (3), äçëáäÞ          
                                                                                      (((d^2) y)/(dt^2))-(1/t)(dy/dt)+4(t^2)y=0.   
Ðáñáôçñïýìå üôé p (t) = -(1/t) .   Õðïëïãßæïõìå ðñþôá ôçí ïñßæïõóá Wronski,  ìå ôïí ôýðï ôïõ Abel:

W = C*Exp[-Integrate[-(1/t), t]]

y1 = Sin[(t^2)]

y2 = y1 ∫(W/(y1^2)) \[DifferentialD]t

EðïìÝíùò ç ãåíéêÞ ëýóç ôçò åîßóùóçò åßíáé ç y (t) = (Subscript[C, 1]) sin ((t^2)) - (1/2) 
              (Subscript[C, 2]) cos ((t^2)) .   ÓxåäéÜæïõìå ôç ãñáöéêÞ ðáñÜóôáóç ôçò ëýóçò, ãéá ôéò ôéìÝò ôùí óôáèåñþí (Subscript[C, 1])=2 êáé (Subscript[C, 2]) = 3.

f[x_] := 2  Sin[(x^2)] - (3/2)  Cos[(x^2)]

Plot[f[x], {x, -7, 7}, 
      PlotStyle -> {RGBColor[0, 0, 1]}]

2.1.  ÃñáììéêÝò äéáöïñéêÝò åîéóþóåéò 2çò ôÜîçò ìå óôáèåñïýò óõíôåëåóôÝò êáé åöáñìïãÝò.

Óôçí åíüôçôá áõôÞ, óõíåxßæïõìå ôç ìåëÝôç ôùí ãñáììáéêþí äéáöïñéêþí åîéóþóåùí 2çò ôÜîçò .   Èá áóxïëçèïýìå ìå ôçí åðßëõóç åîéóþóåùí ìå óôáèåñïýò óõíôåëåóôÝò êáé èá óõæçôÞóïõìå ïñéóìÝíåò åöáñìïãÝò.

Á.   Ç ïìïãåíÞò åîßóùóç
H ïìïãåíÞò ãñáììéêÞ äéáöïñéêÞ åîßóùóç 2  çò ôÜîçò ìå óôáõåñïýò óõíôåëåóôÝò,  åßíáé ç 

                    a(((d^2) y)/(dt^2))+b  (dy/dt)+cy=0,  a,b,c∈ℝ, a≠0.         (1)
Ãéá ôçí åýñåóç ôùí ëýóåùí,  èá xñçóéìïðïéÞóïõìå ôç ëåãüìåíç äïêéìáóôéêÞ ëýóç		
     
                                         y = (E^(r t))                                                                  (2)

üðïõ ï áñéèìüò    r, èá ðñïóäéïñéóôåß óôç óõíÝxåéá.

Ðáñáôçñþíôáò áðü ôç  (2)    üôé

                           y' = r(E^(r t))         êáé      y'' = (r^2) (E^(r t))                            (3)  
              
êáé áíôéêáèéóôþíôáò ôéò åêöñÜóåéò  (2) êáé (3) óôçí åîßóùóç (1) ëáìâÜíïõìå ôç óxÝóç

           0 = ay'' + by' + 
              cy = ((ar^2)  + br + c) (E^(r t))  = 
              0  .         (4)   

Ãéá íá éóxýåé ç åîßóùóç (4) åßíáé öáíåñü üôé èá ðñÝðåé 

                                      (ar^2)  + br + c = 
        0   .                                            (5)

Ç åîßóùóç (5) ëÝãåôáé xáñáêôçñéóôéêÞ åîßóùóç ôçò äéáöïñéêÞò åîßóùóçò (1) .  Ç öýóç ôùí ëýóåùí ôçò äéáöïñéêÞò åîßóùóçò  (1),åîáñôÜôáé áðü ôï áí ç åîßóùóç (5) Ýxåé äéáêåêñéìÝíåò ðñáãìáôéêÝò ñßæåò, ìéá ñßæá (ðïëëáðëüôçôáò 2) Þ óõæõãåßò ìéãáäéêÝò ñßæåò.

Á  .1  ÐñáãìáôéêÝò êáé äéáêåêñéìÝíåò ñßæåò

ÐáñÜäåéãìá 1: Âñåßôå Ýíá èåìåëéþäåò óýíïëï ëýóåùí ãéá ôçí åîßóùóç 
                                     y'' - 4  y' + 3  y=0    .         
Ëýóç: Óýìöùíá ìå ôá ðáñáðÜíù, ç xáñáêôçñéóôéêÞ åîßóùóç åßíáé ç 

                 (r^2) - 4  r + 5 = (r - 3) (r - 1) = 0.  

EðïìÝíùò ïé äýï ëýóåéò ôçò åîßóùóçò äßíïíôáé áðü ôéò

                         (Subscript[y, 1])=(e^(3  t))    êáé  (Subscript[y, 1])=(e^t)   .

Ïé ëýóåéò áõôÝò, ðñÜãìáôé óõíéóôïýí Ýíá èåìåëéþäåò óýíïëï ëýóåùí .  
              Aêïëïõèþíôáò ôç óõæÞôçóç  ãéá ôçí ïñßæïõóá Wronski (âë .  
                  2.1  .2) ðáñáôçñïýìå üôé 

              W ( (Subscript[y, 1]), (Subscript[y, 2])) (t) =det(({{(Subscript[y, 1]), (Subscript[y, 2])}, {((Subscript[y, 1])^′), ((Subscript[y, 2])^′)}}))(t)= 2(E^(4  t))≠0.

'Áñá ðñÜãìáôé óõíéóôïýí èåìåëéþäåò óýíïëï ëýóåùí .  ÅðïìÝíùò ç ãåíéêÞ ëýóç Ýéíáé ç

                                          y = (Subscript[C, 1]) (e^(3  t)) + (Subscript[C, 2]) (e^t) ,

üðïõ   (Subscript[C, 1]) êáé (Subscript[C, 2])åßíáé áõèáßñåôåò óôáèåñÝò.

ÁÓÊÇÓÇ 1 :  Íá áðïäåßîåôå üôé  óôç ãåíéêÞ ðåñßðôùóç,  ôùí 2 äéáêåêñéìÝíùí ñéæþí ôçò xáñáêôçñéóôéêÞò åîßóùóçò (5) Ýóôù (Subscript[r, 1]) êáé  (Subscript[r, 2,  ]) ïé ëýóåéò  
                                       
                                             (Subscript[y, 1]) = (e^((Subscript[r, 1]) t))    êáé  (Subscript[y, 2]) = (e^((Subscript[r, 2]) t)) 

óõíéóôïýí èåìåëéþäåò óýíïëï ëýóåùí.

ÐáñÜäåéãìá 2:  Íá âñåßôå ôç ëýóç ôçò åîßóùóçò 
                                    
                                                   y'' - 4  y' + 3  y = 0,

ç ïðïßá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò y(0)=1 êáé y' (0) = 5. 

Ëýóç:  Xñçóéìïðïéïýìå ôçí Ýêöñáóç ôçò ãåíéêÞò ëýóçò ðïõ âñÞêáìå áðü ôï ÐáñÜäåéãìá 1,

                                                      y (t)=(Subscript[C, 1]) (e^(3  t))+(Subscript[C, 2])(e^t).              (6)

H (6) ðñÝðåé íá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò  :  Ðñïêýðôåé ôï óýóôçìá ôùí åîéóþóåùí
      
                                                                     {({{(Subscript[C, 1]) + (Subscript[C, 2]) = 1}, {3  (Subscript[C, 1]) + (Subscript[C, 2]) = 5. }})

Ç ëýóç ôïõ ðáñáðÜíù óõóôÞìáôïò åßíáé
 (Subscript[C, 1])=2 êáé (Subscript[C, 2])=-1.     ÅðïìÝíùò ç ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí óõíèçêþí åßíáé ç  y(t)=2  (e^(3  t)) - (e^t).

Ãéá ìéá áêüìç öïñÜ èá xñçóéìïðïéÞóïõìå ôçí åíôïëÞ ôïõ  MATHEMATICA   DSolve :

DSolve[y''[t] - 4  y'[t] + 3  y[t] == 
        0, 
y[t], t]

ç ïðïßá äßíåé Üìåóá ôç ãåíéêÞ  ëýóç ôçò åîßóùóçò  ôïõ Ðáñáäåßãìáôïò 1.  Ãéá íá ëýóïõìå ôï ðñüâëçìá áñxéêþí óõíèçêþí ãñÜöïõìå ôçí åíôïëÞ ìå ôïí áêüëïõèï ôñüðï,  ëáìâÜíïíôáò õðüøéí ôéò áñxéêÝò óõíèÞêåò

s=DSolve[{y''[t]-4y'[t]+3  y[t]==0,y[0] == 1,y'[0]==5},y[t],t]

Ç ãñáöéêÞ ðáñÜóôáóç ôçò ëýóçò:

Plot[
      y[t] /. s[[1]], {t, -1, 
        1}, 
PlotStyle -> {RGBColor[1, 0, 
            0]}]

Á  .2  Ðåñßðôùóç äéðëÞò ñßæáò
Èåùñïýìå ôþñá ôçí ðåñßðôùóç,  üðïõ ç xáñáêôçñéóôéêÞ  åîßóùóç Ýxåé ìéá äéðëÞ ñßæá ( ðïëëáðëüôçôáò 2) .  Óå áõôÞ ôçí ðåñßðôùóç ç ñßæá ôçò xáñáêôçñéóôéêÞò åîßóùóçò åßíáé ç  
                                                                                
                                                                                  r=-(b/(2  a)).

Eßíáé öáíåñü üôé ìéá ëýóç ôçò åîßóùóçò  åßíáé ç 
                                                                  
                                                                                  (Subscript[y, 1]) (t)=(E^rt) .

Méá äåýôåñç ëýóç èá ðñïêýøåé áðü ôç ìÝèïäï õðïâéâáóìïý ôçò ôÜîçò (2.1  .3) .  ÅðáíåîåôÜæïíôáò ôïí ôýðï ôïõ Ábel, âëÝðïõìå üôé ç ïñßæïõóá Wronski äýï ïðïéïíäÞðïôå ëýóåùí ôçò äéáöïñéêÞò åîßóùóçò
                                                                                      
                                                                      y'' + p (t) y' + q (t) y = 0,

éêáíïðïéåß ôç äéáöïñéêÞ åîßóùóç  W' + 
                p (t) W = 0.  
Óôçí ðåñßðôùóç ìáò  (åîßóùóç ìå óôáèåñïýò óõíôåëåóôÝò) ç ïñßæïõóá  Wronski éêáíïðïéåß ôçí åîßóùóç

                                                                                         W' + (b/a) W = 0, 
ç ïðïßá Ýxåé ëõóç  ôçí

                                                                         W (t) = C(E^(-(bt/a))) = 
                C(E^(2  rt)),

üðïõ C  åßíáé ìéá áõèáßñåôç óôáèåñÜ .  Ôþñá ç ìÝèïäïò õðïâéâáóìïý ôçò ôÜîçò äßíåé ôç äåýôåñç ëýóç .  Áðü ôï ïëïêëÞñùìá 

                                                                   ((Subscript[y, 1])/(Subscript[y, 2]))=∫((W (t))/((Subscript[y, 1]) (((t))^2))) \[DifferentialD]t=∫C((E^(2  rt))/(E^(2  rt))) \[DifferentialD]t=Ct.

Bñßóêïõìå ëïéðüí üôé ç  äåýôåñç ëýóç  åßíáé Þ 

                                                                                     (Subscript[y, 2]) (t)=Ct(E^rt) .

ÁÓÊÇÓÇ 2: Íá áðïäåßîåôå üôé  óôç ãåíéêÞ ðåñßðôùóç,  ôçò äéðëÞò ñßæáò  ãéá ôç  xáñáêôçñéóôéêÞ  åîßóùóç (5) Ýóôù r,   ïé ëýóåéò  
                                       
                                                        (Subscript[y, 1]) = (e^rt)    êáé  (Subscript[y, 2]) = 
          (te^rt) 

óõíéóôïýí èåìåëéþäåò óýíïëï ëýóåùí.

ÐáñÜäåéãìá 3:  Íá âñåßôå ôç ëýóç ôçò åîßóùóçò 
                                    
                                                   y'' - 4  y' + 4  y = 0,

ç ïðïßá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò y(0)=1 êáé y' (0) = 5. 

Ëýóç:   Ç xáñáêôçñéóôéêÞ åîßóùóç Ýxåé ìéá äéðëÞ ñßæá r=2. 
Xñçóéìïðïéïýìå ôï áðïôÝëåóìá ôçò  ÁÓÊÇÓÇÓ 2.  Ç ãåíéêÞ ëýóç åßíáé,

                                                      y (t)=(Subscript[C, 1]) (e^(2  t))+(Subscript[C, 2])((te)^t).              (7)

H (7)  ðñÝðåé íá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò  :  Ðñïêýðôåé ôï óýóôçìá ôùí åîéóþóåùí
      
                                                                     {({{(Subscript[C, 1]) = 1}, {2  (Subscript[C, 1]) + (Subscript[C, 2]) = 5. }})

Ç ëýóç ôïõ ðáñáðÜíù óõóôÞìáôïò åßíáé
 (Subscript[C, 1]) = 1 êáé (Subscript[C, 2]) = 5.     ÅðïìÝíùò ç ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí óõíèçêþí åßíáé ç  y (t) = 
            (e^(2  t)) + 3  (te^(2  t)).

Ãéá  ôï ðáñÜäåéãìá áõôü çåíôïëÞ ôïõ  MATHEMATICA   DSolve  äßíåé  :

DSolve[y''[t] - 4  y'[t] + 4  y[t] == 
        0, 
y[t], t]

Ç  ãåíéêÞ ëýóç ðïõ ìáò äßíåé ç åíôïëÞ DSolve åßíáé ðñÜãìáôé óùóôÞ .  Âñßóêïõìå ôþñá ôç ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí óõíèçêþí.

q1=DSolve[{y''[t]-4y'[t]+4  y[t]==0,y[0] == 1,y'[0]==5},y[t],t]

Ãéá ôç ãñáöéêÞ ðáñÜóôáóç ôçò ëýóçò Ýxïõìå:

Plot[
      y[t] /. q1[[1]], {t, -0.7, 
        0.7}, 
PlotStyle -> {RGBColor[0, 0, 
            1]}]

Á .3 Ðåñßðôùóç  ìéãáäéêþí  ñéæþí
Èåùñïýìå ôþñá ôçí ðåñßðôùóç,  üðïõ ç xáñáêôçñéóôéêÞ  åîßóùóç Ýxåé óõæõãåßò ìéãáäéêÝò ñßæåò 

                               r = λ ± 
          iμ,   λ = -(b/(2  
                a)),  μ = ((Sqrt[4  ac - (b^2)])/(2  a)),   D = 
        (b^2) - 4  ac ≺ 0. 

Áò áíáæçôÞóïõìå ëýóç ôçò ìïñöÞò

                                                                                             y = (e^λ)t z,

üðïõ z åßíáé ìéá óõíÜñôçóç ôïõ t .   Óêïðåýïõìå íá ðñïóäéïñßóïõìå  ìå áíôéêáôÜóôáóç, xñçóéìïðïéþíôáò ôéò ðáñáãþãïõò ôçò y,
 
                                                                 y' = (e^λ)t z' + λ(e^λ)t z
                                                                 y''=(e^λ)t z''+2  λ(e^λ)t z'+(λ^2)(e^λ)tz.

Áðü ôç åîßóùóç (1), âñßóêïõìå 
             
                                               ay'' + by' + cy   =  [a (z'' + 2  λz' + (λ^2) z) + 
                    b (z' + λz) + cz] (e^λ)t
                                                                            =  [az'' + (2  aλ + b) 
                        z' + (a(λ^2) + bλ + c) z](e^λ)t.

Må ôçí åðéëïãÞ  λ=-(b/(2  a)),  âñßóêïõìå üôé ç Üãíùóôç óõíÜñôçóç z, ðñÝðåé íá áðïôåëåß ëýóç ôçò äéáöïñéêÞò åîßóùóçò 2  çò ôÜîçò

                                                                                                  z'' + (μ^2) z = 0,

ôçò ïðïßáò ç ãåíéêÞ ëýóç (âë .  
                  2.  ÐáñÜäåéãìá 2) äßíåôáé áðü ôç óxÝóç

                                                                              z= (Subscript[C, 1]) cos (μt)+(Subscript[C, 2]) sin (μt).

ÁÓÊÇÓÇ 3: Íá áðïäåßîåôå üôé  óôç ãåíéêÞ ðåñßðôùóç,  ô  ùí  óõæõãþí  ìéãáäéêþí  ñéæþí   ãéá  ôç   
              xáñáêôçñéóôéêÞ   åîßóùóç  (5)  Ýóôù  r=λ ± iμ,   ïé ëýóåéò  
                                       
                                             (Subscript[y, 1])=(e^λ)t  cos(μt)  êáé  (Subscript[y, 2])=(e^λ)tsin(μt) 

óõíéóôïýí èåìåëéþäåò óýíïëï ëýóåùí.

ÐáñÜäåéãìá 4:  Íá âñåßôå ôç ëýóç ôçò åîßóùóçò 
                                    
                                                   y'' + y' + 1.25  y = 0,

ç ïðïßá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò y(0)=1 êáé y' (0) = 0. 

Ëýóç:   Ç xáñáêôçñéóôéêÞ åîßóùóç Ýxåé óõæõãåßò ìéãáäéêÝò ñßæåò  r=-0.5±i. 
Xñçóéìïðïéïýìå ôï áðïôÝëåóìá ôçò  ÁÓÊÇÓÇÓ 3. Ç ãåíéêÞ ëýóç åßíáé,

                                                      y (t) = (Subscript[C, 1]) (e^(-0.5 t)) 
                cost + (Subscript[C, 2]) (e^(-0.5 t)) 
                sint              (8)

H (8)  ðñÝðåé íá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò  :  Ðñïêýðôåé ôï óýóôçìá ôùí åîéóþóåùí
      
                                                                     {({{y (0) = (Subscript[C, 1]) = 1}, { y' (0) = -0.5 (Subscript[C, 1]) + (Subscript[C, 2])}})

Ç ëýóç ôïõ ðáñáðÜíù óõóôÞìáôïò åßíáé
 (Subscript[C, 1])=1 êáé (Subscript[C, 2])=0.5 .     ÅðïìÝíùò ç ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí óõíèçêþí åßíáé ç  y (t)=(e^(-0.5 t))(cost + 0.5  sint).

Xñçóéìïðïéþíôáò ôï  MATHEMATICA:

DSolve[y''[t] + y'[t] + 1.25  y[t] == 
        0, 
y[t], t]

Âñßóêïõìå ôþñá ôç ëýóç ôïõ ðñïâëÞìáôïò áñxéêþí óõíèçêþí.

q2=DSolve[{y''[t]+y'[t]+1.25*y[t]==0,y[0] == 1,y'[0]==0},y[t],t]

Plot[
      y[t] /. q2[[1]], {t, -8, 
        25}, 
PlotStyle -> {RGBColor[0, 0, 
            1]}]

ÁÓÊÇÓÇ 4:   Xñçóéìïðïéþíôáò ôï Ìathematica, åðéëýóôå ôï ðñüâëçìá áñxéêþí óõíèçêþí ôïõ Ðáñáäåßãìáôïò  4, ãéá äéÜöïñåò áñxéêÝò óõíèÞêåò êáé óxåäéÜóôå ôç ëýóç .   Ðùò óõìðåñéöÝñïíôáé ïé ëýóåéò  êáèþò  ôï  t   áõîÜíåé (t->+∞); Ôé äéáöïñÝò ðáñáôçñåßôáé óôéò ãñáöéêÝò ðáñáóôÜóåéò .  
              Xñçóéìïðïéþíôáò ôçí Ýêöñáóç ôçò ãåíéêÞò ëýóçò, ìðïñåßôå íá ôåêìçñéþóåôå ôéò ðáñáôçñÞóåéò óáò;

Á.  ÃåíéêÞ ÌÝèïäïò åðßëõóçò ãéá ôçí ïìïãåíÞ ãñáììéêÞ äéáöïñéêÞ åîßóùóç 2çò ôÜîçò ìå  óôáèåñïýò 
óõíôåëåóôÝò.


        ÊáôáóêåõÜæïõìå ôçí xáñáêôçñéóôéêÞ  åîßóùóç  áðü ôçí åîßóùóç  (1)                                                                                 
                                                                 
                                                                           (ar^2)  + br + c = 0  ,  a ≠ 0.          

      ÐÅÑÉÐÔÙÓÇ 1  (2 äéáêåêñéìÝíåò ñßæåò  (Subscript[r, 1]) êáé (Subscript[r, 2]))
                    H ãåíéêÞ ëýóç ôçò åîßóùóçò  (1) åßíáé   
                                                                                        
                                                                 y (t) =  (Subscript[C, 1]) (e^((Subscript[r, 1]) t)) + (Subscript[C, 2]) 
                (e^((Subscript[r, 2]) t))   ,       (Subscript[C, 1]) ,(Subscript[C, 2])  óôáèåñÝò. 

  ÐÅÑÉÐÔÙÓÇ 2  (1 äéðëÞ ñßæá  r) 

                    H ãåíéêÞ ëýóç ôçò åîßóùóçò  (1) åßíáé   
                                                                                        
                                                                 y (t) =  (Subscript[C, 1]) (e^rt) + (Subscript[C, 2]) (te^rt)   ,       (Subscript[C, 1]) ,(Subscript[C, 2])  óôáèåñÝò. 

 ÐÅÑÉÐÔÙÓÇ 3  (óõæõãåßò ìéãáäéêÝò ñßæåò  r=λ ± iμ) 

                    H ãåíéêÞ ëýóç ôçò åîßóùóçò  (1) åßíáé   
                                                                                        
                                                                 y (t) =  (Subscript[C, 1]) (e^λ)t cos (μt) + (Subscript[C, 2]) 
            (e^λ)t  sin (μt) ,        (Subscript[C, 1]) , 
      (Subscript[C, 2])  óôáèåñÝò.

Â .  Ç ìç ïìïãåíÞò åîßóùóç 

H  ìç ïìïãåíÞò ãñáììéêÞ äéáöïñéêÞ åîßóùóç 2  çò ôÜîçò ìå óôáèåñïýò óõíôåëåóôÝò,  åßíáé ç 

                         a(((d^2) y)/(dt^2))+b  (dy/dt)+cy=g (t),  a,b,c∈ℝ, a≠0.         (1)

H óõíÜñôçóç g(t) ïíïìÜæåôáé óõíÜñôçóç äéÝãåñóçò .  

Ãéá ôçí åîßóùóç (1) ðáñÝxïíôáé äýï ìÝèïäïé åðßëõóçò .   Ç ìÝèïäïò ôùí áðñïóäéïñßóôùí óõíôåëåóôþí êáé ç
 ìÝèïäïò ìåôáâïëÞò ôùí ðáñáìÝôñùí.  Óôéò óçìåéþóåéò áõôÝò èá ðåñéïñéóôïýìå óôç óõíïðôéêÞ ðáñïõóéÜóç ôçò äåýôåñçò  .   Ìïëïíüôé ìåéïíåêôåß ùò ðñïò ôçí ðñþôç üóïí áöïñÜ ôçí ðïóüôçôá ôùí õðïëïãéóìþí ðïõ ðñÝðåé íá äéåêðáéñåùèïýí ãéá ôçí Ýõñåóç ôçò ëýóçò, õðåñôåñåß ùò ðñïò  Ýíá ðïëý óçìáíôéêü ðëåïíÝêôçìá .   Óå áíôßèåóç ìå ôçí ðñþôç,ç ïðïßá áíôéìåôùðßæåé ïñéóìÝíåò ðåñéðôþóåéò åîùôåñéêþí äéåãÝñóåùí (ðïëõùíõìéêÝò,  åêèåôéêÝò,  ôñéãùíïìåôñéêÝò Þ óõíäõáóìü áõôþí) , ç ìÝèïäïò ðáñåìâïëÞò ôùí ðáñáìÝôñùí ìáò äßíåé ðÜíôá ôç ëýóç áíåîÜñôçôá áðü ôï ôç ìïñöÞ ôçò åîùôåñéêÞò äéÝãåñóçò!

Â1 .  Ç ìÝèïäïò ìåôáâïëÞò ôùí ðáñáìÝôñùí
                                                        
Èá ðåñéãñÜøïõìå ôç ìÝèïäï ãéá ôç ãåíéêÞ ãñáììéêÞ äéáöïñéêÞ åîßóùóç 2  çò ôÜîçò, 

                                         (( (d^2) y)/(dt^2)) + p (t) (dy/dt) + 
              q (t) y = g (t),                         (2)

(äçëáäÞ ç ìÝèïäïò åöáñìüæåôáé êáé óôçí ðåñßðôùóç ôùí ìç óôáèåñþí óõíôåëåóôþí!). 

ÎåêéíÜìå ðñþôá áðü ôçí ïìïãåíÞ åîßóùóç 

                                               (( (d^2) y)/(dt^2)) + p (t) (dy/dt) + 
              q (t) y = 0,                         (3)

ç ïðïßá õðïèÝôïõìå üôé Ýxåé ôç ãåíéêÞ ëýóç

                                      y (t) = (Subscript[C, 1]) (Subscript[y, 1]) (t) + (Subscript[C, 2]) (Subscript[y, 2]) (t) .                                      (4)

Öáßíåôáé ëïãéêü íá áíáæçôÞóïõìå ìéá åéäéêÞ ëýóç ãéá ôçí (2)   ôçò ìïñöÞò

                          (Subscript[y, ϵ]) (t) = (Subscript[u, 1]) (t) (Subscript[y, 1]) (t) + (Subscript[u, 2]) (t) (Subscript[y, 2]) (t) .                                (5)

üðïõ ïé óôáèåñÝò (Subscript[C, 1]),  (Subscript[C, 2,  ]) Ýxïõí áíôéêáôáóôáèåß,  áðü (ðéèáíüí) ìç óôáèåñÝò óõíáñôÞóåéò (Subscript[u, 1]) (t), (Subscript[u, 2]) (t) .  Ðáñáôçñïýìå üôé 

\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(y\_ϵ\%'\) \((t)\) = \(u\_1\) \(y\_1\%'\)\  + \(u\_1\%'\) 
            y\_1\  + \ \(u\_2\) \(y\_2\%'\)\  + \(u\_2\%'\) 
            y\_\(\(2\)\(\ \)\(.\)\(
\)\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((6)\)\)\)
\(\(H\ äåýôåñç\ ðáñÜãùãïò\ \ \(y\_ϵ\%''\) \((t)\), \ èá\ ðåñéëáìâÜíåé\ ïêôþ\ \(üñïõò!!\)\ \ \ Ãéá\ íá\ Ýxoõìå\ áðëïýóôåñåò\ åêöñÜóåéò, \ èá\ áðáéôÞóïõìå\)\(
\)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(u\_1\%'\) y\_1\  + \ \(u\_2\%'\) 
            y\_\(\(2\)\(\ \)\(
\)\) = 
        0. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((7)\)\)\)
Ìå áõôü ôïí ôñüðï, 
        xñçóéìïðïéþíôáò ôéò  (6),  (7)  Ýxoõìå

\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(y\_ϵ\%'\) \((t)\) = \ \(u\_1\) \(y\_1\%'\)\  + \ \ \(u\_2\) \(y\_\(\(2\)\(\ \)\(,\)\)\%'\)\)\)
\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(y\_ϵ\%''\) \((t)\)\  = \ \(u\_1\) \(y\_1\%''\)\  + \(u\_1\%'\) \(y\_1\%'\)\  + \ \(u\_2\) \(y\_2\%''\)\  + \ \(u\_2\%'\) \(y\_2\%'\) . 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
Ìå\ áíôéêáôÜóôáóç\ óôç\ \((2)\), \ ëáìâÜíïõìå\)\(\ \)\(
\)\)
\(\(\ \)\(\(y\_ϵ\%''\) \((t)\)\  + 
          p \((t)\) \(y\_ϵ\%'\) \((t)\) + 
          q \((t)\) \(y\_ϵ\) \((t)\) = \(u\_1\) \((\(y\_1\%''\) \((t)\)\  + p \((t)\) \(y\_1\%'\) \((t)\) + 
                q \((t)\) \(y\_1\) \((t)\))\) + \ \(u\_1\%'\) \(y\_1\%'\)\ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  + \(u\_2\) \((\(y\_2\%''\) \((t)\)\  + p \((t)\) \(y\_2\%'\) \((t)\) + 
                q \((t)\) \(y\_2\) \((t)\))\) + \(u\_2\%'\) \(y\_2\%'\)\)\(\ \ \ \ \)\(
\)\)
Ç ôåëåõôáßá ðáñÜóôáóç  áðëïðïéåßôáé åðéðëÝïí,  áöïý ïé (Subscript[y, 1]), (Subscript[y, 2 ]) åßíáé ëýóåéò ôçò ïìïãåíïýò åîßóùóçò  (3) .  ' Åxïõìå

\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(y\_ϵ\%''\) \((t)\)\  + 
          p \((t)\) \(y\_ϵ\%'\) \((t)\) + 
          q \((t)\) \(y\_ϵ\) \((t)\) = \ \(u\_1\%'\) \(y\_1\%'\)\  + \ \(u\_2\%'\) \(y\_2\%'\) . \ \ \ \ \ \((9)\)\)\(
\)\)
Xñçóéìïðïéþíôáò ôþñá ôéò   (2),  (7), (9) (õðåíèõìßæïõìå üôé ç (Subscript[y, ϵ])   åßíáé ëýóç ôçò (2)),  ëáìâÜíïõìå ôï óýóôçìá ôùí åîéóþóåùí

                                     ({{\(\(\(u\_1\%'\) y\_1\  + \ \(u\_2\%'\) 
                          y\_\(\(2\)\(\ \)\(
\)\) = 
                      0\)\(,\)\)}, {\(\(\ \)\(\(u\_1\%'\) \(y\_1\%'\)\  + \ \(u\_2\%'\) \(y\_2\%'\) = g \(\((t)\)\(.\)\)\)\)}})                                             (10)

Ðáñáôçñåßóôå üôé ç ïñßæïõóá ôùí óõíôåëåóôþí ôïõ óõóôÞìáôïò (10) åßíáé ç ïñßæïõóá Wronski! Eðéëýïõìå ôï óýóôçìá ìå ôïí êáíüíá ôïõ Cramer,  êáé âñßóêïõìå ôç ëýóç

                                \(u\_1\%'\)=((det(({{0, (Subscript[y, 2])}, {g (t), \(y\_2\%'\)}})))/(det(({{(Subscript[y, 1]), (Subscript[y, 2])}, {\(y\_1\%'\), \(y\_2\%'\)}}))))=-(((Subscript[y, 2]) 
                      g (t))/(W ((Subscript[y, 1]), (Subscript[y, 2])) (t))),       (11) 

                               \(u\_1\%'\)=((det(({{(Subscript[y, 1]), 0}, {\(y\_1\%'\), g (t)}})))/(det(({{(Subscript[y, 1]), (Subscript[y, 2])}, {\(y\_1\%'\), \(y\_2\%'\)}}))))=(((Subscript[y, 1]) 
                    g (t))/(W ((Subscript[y, 1]), 
                      (Subscript[y, 2])) (t))) .            (12)

H ïëïêëÞñùóç ôùí (11),  (12) ìáò äéíåé

                  (Subscript[u, 1]) (t) = -∫(((Subscript[y, 2]) (s) 
                      g (s))/(W ((Subscript[y, 1]), 
                        (Subscript[y, 2])) (s))) \[DifferentialD]s    (Subscript[\[VerticalSeparator] , s -> 
                t

])                              (13)

                (Subscript[u, 2]) (t)  =    ∫(((Subscript[y, 1]) (s) 
                    g (s))/(W ((Subscript[y, 1]), 
                      (Subscript[y, 2])) (s))) \[DifferentialD]s    (Subscript[\[VerticalSeparator] , s -> 
                t

])                               (14)
Xñçóéìïðïéþíôáò ôþñá ôçí (5) êáé ôéò (13),  (14),  âñßóêïõìå üôé

      (Subscript[y, ϵ]) (t) = ∫(((Subscript[y, 1]) (s) (Subscript[y, 2]) (t) - (Subscript[y, 1]) (t) (Subscript[y, 2]) (s))/(W ((Subscript[y, 1]), (Subscript[y, 2])) (s))) 
              g (s) \[DifferentialD]s   (Subscript[\[VerticalSeparator] , s -> t

])  .      (15)

Â1 .  ÌåôáâïëÞ ôùí ðáñáìÝôñùí

         BHMA 1.      ÌåôáôñÝðïõìå ôçí åîßóùóç                                                                                   
                                                                                a(t)(((d^2) y)/(dt^2))+b (t) (dy/dt)+c (t) y=g (t),  a(t)≠0 ,       (i)
                                     óôçí ìïñöÞ ôçò åîßóùóçò
                                                                                      (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y=g(t).                               (ii)           

        
         ÂÇÌÁ 2.      Âñßóêïõìå  Ýíá èåìåëéþäåò óýóôçìá ëýóåùí  (Subscript[y, 1])(t), (Subscript[y, 2])(t)  ôçò ïìïãåíïýò åîßóùóçò  
                                                                                      (((d^2) y)/(dt^2)) + p (t) (dy/dt) + q (t) y = 
          0.    

         
        ÂÇÌÁ 3.     H åéäéêÞ ëýóç ôçò åîßóùóçò (i)  äßíåôáé áðü ôç óxÝóç

                                                            (Subscript[y, ϵ]) (t)=∫(((Subscript[y, 1]) (s) (Subscript[y, 2]) (t) - (Subscript[y, 1]) (t) (Subscript[y, 2]) (s))/(W ((Subscript[y, 1]), (Subscript[y, 2])) (s))) 
                g (s) \[DifferentialD]s   (Subscript[\[VerticalSeparator] , s -> 
                  t

])  .

ÐáñÜäåéãìá 5: Âñåßôå ôç ãåíéêÞ ëýóç ôçò åîßóùóçò   
                                                                                  (((d^2) y)/(dt^2))+y =cot (t)    (a)     
Ëýóç:   Ðáñáôçñïýìå üôé  ç ìïñöÞ ôçò åîùôåñéêÞò äéÝãåñóçò, äåí åìðßðôåé óôéò ìïñöÝò ðïõ êáèéóôïýí äõíáôÞ ôçí åöáñìïãÞ ôçò ìåèüäïõ ôùí áðñïóäéïñßóôùí óõíôåëåóôþí .  Ðáñáôçñïýìå üôé ïé óõíáñôÞóåéò   
(Subscript[y, 1]) (t) = cost,  (Subscript[y, 2]) (t) = sint, áðïôåëïýí Ýíá èåìåëéþäåò óýíïëï ëýóåùí.   Ãíùñßæïíôáò ôç ìÝèïäï åðßëõóçò äåí åßíáé äýóêïëï íá  ãñÜøïõìå  Ýíá "ðñüãñáììá" Mathematica ãéá ôçí åöáñìïãÞ ôçò ìåèüäïõ!!   To ðñüãñáììá ìáò äåí åßíáé ôßðïôá  Üëëï ðåñáí áðü ôçí õëïðïßçóç ôùí óxÝóåùí (13), (14) êáé (15) ìÝóù ôïõ Mathematica

MetavolhParametron[y1_, y2_, g_][
        t_] := 
Module[{Wronskian, u1, 
          u2}, 
Wronskian = 
          Simplify[
            y1[t] y2'[t] - y2[t] y1'[t]]; 
u1 = 
          Simplify[
            Integrate[-y2[t] g[t]/Wronskian, t]]; 
u2 = 
          Simplify[
            Integrate[y1[t] g[t]/Wronskian, t]]; 
Simplify[
          y1[t] u1 + y2[t] u2]]

Ç åíôïëÞ  "Wronskian=..." õðïëïãßæåé ôçí ïñßæïõóá Wronski, ãéá ôéò ëýóåéò (Subscript[y, 1])(t),  (Subscript[y, 2])(t) ôçò ïìïãåíïýò åîßóùóçò .   Ïé åíôïëÝò   "u1,u2=..." äåí åßíáé ðáñÜ ïé ôýðïé (13) êáé (14) .  Ç ôåëåõôáßá åíôïëÞ åßíáé ï ôýðïò  (15) .  Åöáñìüæïõìå ôï ðñüãñáììá  óôï  óõãêåêåñéìÝíï ðáñÜäåéãìá.

MetavolhParametron[Cos,  Sin,  Cot][
      t]

ÅðïìÝíùò ç åéäéêÞ ëýóç ôïõ ðñïâëÞìáôïò åßíáé ç  

                     (Subscript[y, ϵ]) (t) = 
        log (tan (t/2)) 
          sint .   

Ç  ãåíéêÞ  ëýóç  ôçò   (a)   åßíáé  ç  

   y (t)=(Subscript[C, 1])sint +(Subscript[C, 2]) cost +log (tan (t/2)) sint.

Èá óõêñßíïõìå ôç ëýóç ðïõ âñÞêáìå ìå áõôÞ ðïõ èá ìáò äþóåé ôï MATHEMATICA  ìå ôçí åíôïëÞ DSolve:

sol = 
      DSolve[y''[t] + y[t] == Cot[t], 
y[t], 
        t]

Ç ëýóç ðïõ ìáò äßíåé ç åíôïëÞ DSolve åßíáé ðñÜãìáôé óùóôÞ (êÜíôå ôéò áðëïðïéÞóåéò) .  
ÓxåäéÜæïõìå ôþñá ôç ëýóç ãéá äéÜöïñåò ôéìÝò ôùí óôáèåñþí.

p1 = y[t] /. sol[[1]]

Plot [
      Evaluate[
Table[
          p1 /. {C[1] -> i, C[2] -> i}, {i, 1, 3}], {t, 0, 
          3}], 
PlotStyle -> {RGBColor[1, 0, 0], 
          RGBColor[0, 1, 0], 
RGBColor[0, 0, 1], 
          RGBColor[1, 1, 0], RGBColor[0, 1, 1]}]

ÁÓÊÇÓÇ 5:   Xùñßò ôç xñÞóç ôoõ õðïëïãéóôÞ, íá åðéâåâáéþóåôå ôç ëýóç ôïõ ðáñáäåßãìáôïò 5, êÜíïíôáò ôïõò õðïëïãéóìïýò.

ÐáñÜäåéãìá 5: Âñåßôå ôç ãåíéêÞ ëýóç ôçò åîßóùóçò   
                                                                                  (((d^2) y)/(dt^2))+(ω^2)y =cos(ωt)    (b)     
Ëýóç:Èá xñçóéìïðïéÞóïõìå ôï \!\(\*
StyleBox["\"",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ðñïãñáììá",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ðïõ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ãñÜøáìå",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["óôï",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ðñïçãïýìåíï",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ðáñÜäåéãìá",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[".",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)
\!\(\*
StyleBox["'",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["Åíá",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["èåìåëéþäåò",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["óýíïëï",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ëýóåùí",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ãéá",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ôçí",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ïìïãåíÞ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["åîßóùóç",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["äßíåôáé",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
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StyleBox["áðü",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ôéò",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["óõíáñôÞóåéò",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontColor->GrayLevel[0],
FontVariations->{"CompatibilityType"->0}]\)
                                 
                                \!\(\*
StyleBox[\(y\_1\),
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["(",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
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StyleBox["t",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[")",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["=",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["cos",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["(",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["ùt",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[")",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[",",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["  ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[\(y\_2\),
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["(",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["t",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[")",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["=",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["sin",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["(",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ùt",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[")",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[".",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)

\!\(\*
StyleBox["ÅðåéäÞ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ôï",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ðñüãñáììá",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ìáò",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["\"åßíáé ìéá óõíÜñôçóç ôïõ t\"",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["xñåéÜæåôáé",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["íá",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ïñßóïõìå",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["êáôÜëëçëá",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["ôéò",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[" ",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["óõíáñôÞóåéò",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\) \!\(\*
StyleBox[\(y\_1\),
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["(",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
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FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
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StyleBox["t",
FontFamily->"Garamond",
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FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
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"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[")",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox["êáé",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[" ",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False,
"StrikeThrough"->False,
"Masked"->False,
"CompatibilityType"->0,
"RotationAngle"->0}]\)\!\(\*
StyleBox[\(y\_2\),
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["(",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox["t",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[")",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)\!\(\*
StyleBox[".",
"Text",
FontFamily->"Garamond",
FontSize->16,
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}]\)

om1[t_] := Cos[ω*t]

om2[
        t_]:=Sin[ω*t]

Ìå áõôü ôïí ôñüðï ïñßóáìå ôéò ëýóåéò ôçò ïìïãåíïýò .  Ðáñáôçñïýìå üôé ç åîùôåñéêÞ äéÝãåñóç åßíáé ßóç ìå ôçí
 (Subscript[y, 1])(t). Åöáñìüæïõìå ôï ðñüãñáììá ìáò .

MetavolhParametron[om1, om2, om1][t]

ËÜâáìå ôç óõíÜñôçóç

                                                                (Subscript[y, ϵ]) (t)=((cos (ωt) +  ω tsin ( ωt))/(2 (ω^2))), 

ùò åéäéêÞ ëýóç .   Ç óõíÜñôçóç  ((cos (ωt))/(2 (ω^2)))  åßíáé  åðßóçò ëýóç ôçò ïìïãåíïýò .  'Áñá ìéá åéäéêÞ ëýóç ôïõ ðñïâëÞìáôïò åßíáé ç (( tsin ( ωt))/(2 ω)) .  ÅðïìÝíùò ç ãåíéêÞ ëýóç ôïõ (b) åßíáé ç

                                            y (t)=(Subscript[C, 1]) cos(ùt)+(Subscript[C, 2]) sin ( ωt)+ (( tsin ( ωt))/(2 ω)).

Clear["Global`*"]

\(To\ "ðñüãñáììá ðïõ áêïëïèåß, áðïôåëåß ìéá ôñïðïðïßçóç ôïõ " ðñïãñÜììáôïò \*"" 
ôïõ ðáñáäåßãìáôïò 4, êáé äéíåé ôç ãåíéêÞ ëýóç ìéáò ãñáììéêÞò äéáöïñéêÞò åîßóùóçò 2çò ôÜîçò, ìå ôç ìÝèïäï ìåôáâïëÞò ôùí ðáñáìÝôñùí, áí ãíùñßæïõìå Ýíá èåìåëéþäåò óýíïëï ëýóåùí ãéá ôçí ïìïãåíÞ åîßóùóç.
Óçìåéþóôå üôé ç ôåëåõôáßá åíôïëÞ, äßíåé ôçí Ýêöñáóç ôçò ãåíéêÞò ëýóçò ìå ôéò äýï áõèáßñåôåò óôáèåñÝò."\)

TMeP[y1_, y2_, g_][
        t_]:=
Module[{Wronskian, u1, 
          u2}, 
Wronskian = 
          Simplify[
            y1[t] y2'[t] - y2[t] y1'[t]]; 
u1 = 
          Simplify[
            Integrate[-y2[t] g[t]/Wronskian, t]]; 
u2 = 
          Simplify[
            Integrate[y1[t] g[t]/Wronskian, t]]; 
Simplify[
            y1[t] u1 + y2[t] u2] + C[1] y1[t] + 
          C[2] y2[t]]

TMeP[om1, om2, om1][t]

ÓxåäéÜæïõìå ôç ãñáöéêÞ ðáñÜóôáóç ôçò ëýóçò, ãéá äéÜöïñåò ôéìÝò ôùí óôáèåñþí,  êáé ôçò ðáñáìÝôñïõ ù.

lysh = 
      C[1] Cos[t ω] + 
        C[2] Sin[
            t ω] + ((t  Sin[t ω])/(2 ω))

Plot [
      Evaluate[
Table[
          lysh /. { ω -> i, C[1] -> i, C[2] -> i}, {i, 
            1, 3}], {t, -3 π, 
          3  π}], 
PlotStyle -> {RGBColor[1, 0, 
            0], RGBColor[0, 1, 0], 
RGBColor[0, 0, 1], 
          RGBColor[1, 1, 0], RGBColor[0, 1, 1]}]

Clear["Global`*"]

ÐáñÜäåéãìá 6: Âñåßôå ôç  ëýóç ôçò åîßóùóçò
                                                                                  (((d^2) y)/(dt^2))+(ω^2)y =cos(ωt)      

ç ïðïßá éêáíïðïéåß ôéò áñxéêÝò óõíèÞêåò y(0)=ð êáé y' (0) = 
            2  ð, xñçóéìïðïéþíôáò ôï Mathematica.
Ëýóç: Âñßóêïõìå ôç ãåíéêÞ ëýóç ôïõ ðñïâëÞìáôïò, \(ãéáíá\ ôç\ óõãêñßíïõìå\ ìå\ áõôÞ\ ðïõ\ äßíåé\ ôï\ "ðñüãñáììá"\ ðïõ\ ãñÜøáìåóôï\ ðñïçãïýìåíï\ \(\(ðáñÜäåéãìá\)\(.\)\(\ \)\)\)

solg=DSolve[y''[t]+(ω^2)*y[t]==Cos[ω*t],
y[t],t]//
Simplify

Ðáñáôçñïýìå üôé ç ãåíéêÞ ëýóç ðïõ äßíåé ôï Mathematica ìå ôçí åíôïëÞ DSolve,  åßíáé áõôÞ ðïõ áíáìÝíïõìå (áí êÜíïõìå ôéò óxåôéêÝò áðëïðïéÞóåéò êáé èåùñÞóïõìå êáôÜëëçëá ôéò óôáèåñÝò) . 
Åðéëýïõìå ôþñá ôï ðñüâëçìá áñxéêþí óõíèçêþí.

sol2=DSolve[{y''[t]+(ω^2)*y[t]==Cos[ω*t],y[0] == π,y'[0]==2  π},y[t],t]//
Simplify

ÓxåäéÜæïõìå ôç ãñáöéêÞ ðáñÜóôáóç ãéá äéÜöïñåò ôéìÝò ôçò ðáñáìÝôñïõ ù.  Ç ðáñÜìåôñïò ù óôç èåùñßá ôùí ôáëáíôþóåùí, êáëåßôáé ãùíéáêÞ óõxíüôçôá.

sx1 = y[t] /. sol2[[1]]

sx2 = 
      Plot [Evaluate[
Table[
            sx1 /.  ω -> i, {i, 1, 3}], {t, -3 π, 
            3  π}], 
PlotStyle -> {RGBColor[1, 0, 
              0], RGBColor[0, 1, 0], 
RGBColor[0, 0, 1], 
            RGBColor[1, 1, 0], RGBColor[0, 1, 1]}]

ÓxåäéÜæïíôáò ôéò ëýóåéò ãéá ôéìÝò ãùíéáêÞò óõxíüôçôáò óôï äéÜóôçìá [1, 
          3],  íá ðáñÜãïõìå ôçí áêüëïõèç åðéöÜíåéá

sx3 = 
      ListPlot3D[
Evaluate[
          Table[sx1 /.  ω -> i, {i, 1, 3, 
              0.1}, {t, -2 π, 2  π, 0.1}]], 
        ViewPoint -> {2, 2, 
            2}, 
PlotRange -> {-10, 10}, 
        BoxRatios -> {1, 1, 1}, ClipFill -> None]