4.5 Solve non-homogeneous 2nd order ODE, constant coefficients

Problem: Solve

\[ y^{\prime \prime }\left ( t\right ) -1.5y^{\prime }\left ( t\right ) +5y\left ( t\right ) =4\cos \left ( t\right ) \]

with initial conditions

\[ y\left ( 0\right ) =1 , y^{\prime }\left ( 0\right ) =0 \]

To use Matlab ode45, the second order ODE is converted to state space as follows

Given \(y^{\prime \prime }\left ( t\right ) -1.5y^{\prime }\left ( t\right ) +5y\left ( t\right ) =4\cos \left ( t\right ) \), let

\begin{align*} x_{1} & =y\\ x_{2} & =y^{\prime }\\ & =x_{1}^{\prime }\end{align*}

hence

\[ x_{1}^{\prime }=x_{2}\]

and

\begin{align*} x_{2}^{\prime } & =y^{\prime \prime }\\ & =1.5y^{\prime }-5y+4\cos \left ( t\right ) \\ & =1.5x_{2}-5x_{1}+4\cos \left ( t\right ) \end{align*}

Hence we can now write

\[\begin {bmatrix} x_{1}^{\prime }\\ x_{2}^{\prime }\end {bmatrix} =\begin {bmatrix} 0 & 1\\ -5 & 1.5 \end {bmatrix}\begin {bmatrix} x_{1}\\ x_{2}\end {bmatrix} +\begin {bmatrix} 0\\ 4 \end {bmatrix} \cos \left ( t\right ) \]

Mathematica

Remove["Global`*"]; 
eq  = y''[t]-15/10y'[t]+5y[t]==4  Cos[t]; 
ic  = {y'[0]==0,y[0]== 1}; 
sol = First@DSolve[{eq,ic},y[t],t];
 

\[ \frac {1}{5183}( -1704 \sin (t) \sin ^2\left (\frac {\sqrt {71} t}{4}\right )+69 \sqrt {71} e^{3 t/4} \sin \left (\frac {\sqrt {71} t}{4}\right )+4544 \cos (t) \cos ^2\left (\frac {\sqrt {71} t}{4}\right )+639 e^{3 t/4} \cos \left (\frac {\sqrt {71} t}{4}\right )-1704 \sin (t) \cos ^2\left (\frac {\sqrt {71} t}{4}\right )+4544 \sin ^2\left (\frac {\sqrt {71} t}{4}\right ) \cos (t))) \]

Plot[y,{t,0,10}, 
     FrameLabel->{{"y(t)",None}, 
      {"t","Solution"}}, 
     Frame->True, 
     GridLines->Automatic, 
     GridLinesStyle->Automatic, 
     RotateLabel->False, 
     ImageSize->300, 
     AspectRatio->1, 
     PlotRange->All, 
     PlotStyle->{Thick,Red}]
 

Matlab

function e55 
t0 = 0; %initial time 
tf = 10; %final time 
 %initial conditions [y(0)  y'(0)] 
ic =[1 0]'; 
 
[t,y] = ode45(@rhs, [t0 tf], ic); 
 
plot(t,y(:,1),'r') 
title('Solution using ode45'); 
xlabel('time'); 
ylabel('y(t)'); 
grid on 
set(gcf,'Position',[10,10,320,320]); 
 
  function dydt=rhs(t,y) 
    dydt=[y(2) ; 
         -5*y(1)+1.5*y(2)+4*cos(t)]; 
  end 
end