Question

Find the non-trivial solution of the sturn_liouville problem d²y/dx²+λy=0 y(0)=0, y(π)=0

Answer 1

Any eigenvalues of Equation $\dfrac{dy^2}{x^2}+λy=0$ must be positive. If $y$ satisfies the equation with $\lambda>0,$ then

$$y=c_1\cos\sqrt{\lambda}x+c_2\sin\sqrt{\lambda}x$$

where $c_1$ and $c_2$ are constants. The boundary condition $y(0)=0$ implies that $c_1=0.$ Therefore

$$y=c_2\sin\sqrt{\lambda}x$$

The boundary condition $y(\pi) = 0$ implies that

$$c_2\sin\sqrt{\lambda}\pi=0$$

To make $c_2\sin\sqrt{\lambda}\pi=0$ with $c_2 \not= 0,$ we must choose $\sqrt{\lambda} = n,$ where $n$ is a positive integer.

Therefore $\lambda_n^2=n^2$ is an eigenvalue and

$$y_n=\sin(nx)$$

is an associated eigenfunction.