### 4.6 Solve homogeneous 2nd order ODE, constant coeﬃcients (BVP)

Problem: Solve $y^{\prime \prime }\left ( t\right ) +t\ y\left ( t\right ) =0$ with the boundary conditions $y\left ( 0\right ) =3,y\left ( 20\right ) =-1$ For solving with Matlab, the ODE is ﬁrst converted to state space as follows

Given $$y^{\prime \prime }\left ( t\right ) +t\ y\left ( t\right ) =0$$, let \begin {align*} x_{1} & =y\\ x_{2} & =y^{\prime }\\ & =x_{1}^{\prime } \end {align*}

Therefore $x_{1}^{\prime }=x_{2}$ And \begin {align*} x_{2}^{\prime } & =y^{\prime \prime }\\ & =-t\ y\\ & =-t\ x_{1} \end {align*}

This results in $\begin {bmatrix} x_{1}^{\prime }\\ x_{2}^{\prime }\end {bmatrix} =\begin {bmatrix} 0 & 1\\ -t & 0 \end {bmatrix}\begin {bmatrix} x_{1}\\ x_{2}\end {bmatrix}$ Now bvp4c() can be used.

Mathematica

Clear[y,t];
eq  = y''[t]+t y[t]==0;
ic  = {y[0]==3,y[20]==-1};
sol = First@DSolve[{eq,ic},y[t],t];

y   = y[t]/.sol



$\frac {-\sqrt {3} \text {Ai}\left (\sqrt [3]{-1} t\right )+\text {Bi}\left (\sqrt [3]{-1} t\right )-3\ 3^{2/3} \Gamma \left (\frac {2}{3}\right ) \text {Bi}\left (20 \sqrt [3]{-1}\right ) \text {Ai}\left (\sqrt [3]{-1} t\right )+3\ 3^{2/3} \Gamma \left (\frac {2}{3}\right ) \text {Ai}\left (20 \sqrt [3]{-1}\right ) \text {Bi}\left (\sqrt [3]{-1} t\right )}{\sqrt {3} \text {Ai}\left (20 \sqrt [3]{-1}\right )-\text {Bi}\left (20 \sqrt [3]{-1}\right )}$

Plot[Re[y],{t,0,20},
FrameLabel->{{"y(t)",None},
{"t","Solution"}},
Frame->True,
GridLines->Automatic,
GridLinesStyle->Automatic,
RotateLabel->False,
ImageSize->300,
AspectRatio->1,
PlotRange->All,
PlotStyle->{Thick,Red},
Exclusions->None]



Matlab

function e56

t0 = 0; %initial time
tf = 20; %final time

solinit = bvpinit(linspace(t0,tf,5),[3 -1]);

sol = bvp4c(@odefun,@bcfun,solinit);

plot(sol.x(1,:),sol.y(1,:),'r')
title('solution');
xlabel('time');
ylabel('y(t)');
grid;
set(gcf,'Position',[10,10,320,320]);

function dydt=odefun(t,y)
dydt=[y(2); -t.*y(1)];

function res=bcfun(yleft,yright)
res=[yleft(1)-3
yright(1)+1];