Question 11.2.1: Determine the normalized eigenfunctions of the problem (10):......
Determine the normalized eigenfunctions of the problem (10):
y^{\prime\prime}+\lambda y=0,\quad y(0)=0,\quad y(1)=0.
he eigenvalues of this problem are \lambda_{1}=\pi^{2},\lambda_{2}=4\pi^{2},\,\ldots\,,\lambda_{n}=n^{2}\pi^{2},\ \ldots\ , and the corresponding eigenfunctions are k_{1}\sin(\pi x),k_{2}\sin(2\pi x),\,\ldots\,,k_{n}\sin(n\pi x),\,\ldots\,. respectively. In this case the weight function is r(x) = 1. To satisfy equation (20),
\int_{0}^{1}r(x)\phi_{n}^{2}(x)d x=1,\quad n=1,2,\,\dots\,. (20)
we must choose k_{n} so that
\int_{0}^{1}(k_{n}\sin(n\pi x))^{2}d x=1 (23)
for each value of n. Since
k_{n}^{2}\int_{0}^{1}\sin^{2}(n\pi x)d x=k_{n}^{2}\int_{0}^{1}\left({\frac{1}{2}}-{\frac{1}{2}}\cos(2n\pi x)\right)d x={\frac{1}{2}}k_{n}^{2},
Equation (23) is satisfied if k_{n} is chosen to be {\sqrt{2}} for each value of n. Hence the normalized eigenfunctions of the given boundary value problem are
\phi_{n}(x)=\sqrt{2}\sin(n\pi x),\quad n=1,2,3,\,\dots\,. (24)