Question

Solve the boundary value problem Cfrac%7Bdy%7D%7Bdx%7D%20-%205y%3D0" id="TexFormula1" title=" \frac{d^{2}y}{dx^{2}} + 4 \frac{dy}{dx} - 5y=0" alt=" \frac{d^{2}y}{dx^{2}} + 4 \frac{dy}{dx} - 5y=0" align="absmiddle" class="latex-formula">
$y(0)=0$
$y( \pi /2)=1$

Answer 1

Characteristic equation:

$r^2+4r-5=(r+5)(r-1)=0\implies r=-5,r=1$

$\implies y_c=C_1e^{-5x}+C_2e^x$

Given that $y(0)=0$ and $y\left(\dfrac\pi2\right)=1$, you have

$\begin{cases}0=C_1+C_2\\1=C_1e^{-5\pi/2}+C_2e^{\pi/2}\end{cases}\implies C_1=\dfrac{e^{5\pi/2}}{1-e^{3\pi}},C_2=-\dfrac{e^{5\pi/2}}{1-e^{3\pi}}$

so that the particular solution is

$y=\dfrac{e^{5\pi/2}}{1-e^{3\pi}}(e^{-5x}-e^x)$