For the following equation yay0 with boundary conditions y0y

For the following equation y\"+ay=0 with boundary conditions y(0)=y\'(pi)=0 find the eigen values and its corresponding eigen functions.

Solution

Case 1:a=0

y\'\'=0

INtegrating gives

y=Ax+B

y(0)=B=y\'(pi)=A=0

y=0

trivial solution

Case 2:a<0,a=-k^2

y\'\'=k^2y

y=A e^{kx}+B e^{-kx}

y(0)=A+B=0 ie A=-B

y\'=kA( e^{kx}+e^{-kx})

y\'(pi)=0 gives A=0 hence, B=0

so y=0 ie trivial solution

Case 3: a>0,a=k^2

y\'\'=-k^2y

y=A sin(kx)+B cos(kx)

y(0)=B=0

y\'(x)=Ak cos(kx)

y\'(pi)=Ak cos(k pi)=0

HEnce, kpi =(2n+1)pi/2

k=(2n+1)/2,n=0,1,2,3,...