Determine the normalised eigenfunctions for the BVP: $y''+λy=0, y(0)=0, y(1)=0$

Question

Solving it I get: $y(x)=c_1 \cos(x \sqrt{\lambda}) + c_2 \sin (x \sqrt{\lambda})$

$y(0)= C1 + 0 = 0, C1=0$

$y(1)=0+C2\sin(\sqrt{\lambda})=0$

So, $(\sqrt{\lambda})=n\pi$, $({\lambda})=(n\pi)^2$

So, $ψ_n= \sin(nx\pi)$

However, the solution shows that the answer is:

$ψ_n=(\sqrt{2})\sin(nx\pi)$

Where have I gone wrong in this?

Answer 1

The solution is $C_2\sin (n\pi x).$ It is normalized when $$1=\int_{0}^{1} (C_2 \sin(n\pi x))^2 dx=\frac{(C_2)^2}{2}.$$ From this follows $C_2=\sqrt 2.$