Solve the boundary value problem (d^(2)y)/(dx^(2)) + 4 (dy)/(dx) - 5y=0 y(0)=0 y( \pi /2)=1

Question

Solve the boundary value problem
`(d^(2)y)/(dx^(2)) + 4 (dy)/(dx) - 5y=0`


`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(\frac\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=(e^(5\pi/2))/(1-e^(3\pi)),C_2=-(e^(5\pi/2))/(1-e^(3\pi))`

so that the particular solution is


`y=(e^(5\pi/2))/(1-e^(3\pi))(e^(-5x)-e^x)`