Express the function $$f(x) = x(1-x), 0\le x \le 1$$ as an infinite series in terms of the complete set of eigenfunctions $u_{n}(x)$, n$\in \mathbb N$, of the Sturm-Liouville Boundary Value Problem for the following; $ u''(x) + \lambda u(x) = 0$ on the interval $0<x<1$, $u(0) = 0$, $u'(1) = 0$

So far I have tried to find complete set of eigenvalues $\lambda$ and their corresponding eigenfunctions:

A function $u(x)$ can be defined as $u(x) =\sum_{n=1}^{\infty} A_n X_n$ here $X_n$ denotes set of eigenfunctions. $A_n = \frac{\int_{0}^{1} f(x)u_n(x)dx}{\int_{0}^{1} [u_n(x)]^2 dx}$

To find eigenvalues,

case1 : $\lambda >0$

characteristic polynomial for BVP is $r^2 + \lambda = 0$ $r_1, r_2 = +\sqrt{-\lambda},-\sqrt{-\lambda}$

since $\lambda >0$ the general solution to the differential equation is then

$u(x) = C_1 cos(\sqrt{\lambda}x) + C_2sin(\sqrt{\lambda}x)$. Applying the first boundary condition gives 0 = u(0) = $C_1$.

From here, I am kinda stuck any help to this question will be appreciated.

  • $\begingroup$ Now you can differentiate your $u(x) = C_2\sin(\sqrt{\lambda}x)$ with respect to $x$ and find $C_2$ by using second boundary condition. $\endgroup$ – Voliar Oct 25 '16 at 21:28

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