4.4 Solve homogeneous 2nd order ODE with constant coefficients

Problem: Solve

\[ y^{\prime \prime }\left ( t\right ) -1.5y^{\prime }\left ( t\right ) +5y\left ( t\right ) =0 \]

with initial conditions

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

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

Given \(y^{\prime \prime }\left ( t\right ) -1.5y^{\prime }\left ( t\right ) +5y\left ( t\right ) =0\) 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\\ & =1.5x_{2}-5x_{1}\end{align*}

Hence we can now write

\[\begin {bmatrix} x_{1}^{\prime }\\ x_{2}^{\prime }\end {bmatrix} =\begin {bmatrix} 0 & 1\\ -5 & 1.5 \end {bmatrix}\begin {bmatrix} x_{1}\\ x_{2}\end {bmatrix} \]

Now Matlab ODE45 can be used.

Mathematica Remove["Global`*"]; eq = y''[t]-1.5y'[t]+5y[t]==0; ic = {y'[0]==0,y[0]== 1}; sol = First@DSolve[{eq,ic},y[t],t]; y = y[t]/.sol \(1. \left (1. e^{0.75 t} \cos (2.10654 t)-0.356034 e^{0.75 t} \sin (2.10654 t)\right )\) 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 e54 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)]; end end