### 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 `