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Karo-lina-s [1.5K]
3 years ago
5

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
Mathematics
1 answer:
Sonja [21]3 years ago
6 0
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)
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