hello everyone!(adsbygoogle = window.adsbygoogle || []).push({});

I had a stuck in solving problem for a week now, so need help.

Please help!

the problem is as follows.

In a closed interval [itex]I=[0,\pi][/itex], the 2-times continuously differentiable function [itex]\phi(x)[/itex] and [itex]\psi(x)[/itex] meet the following conditions (they're ranged in [itex]\mathbb{R}[/itex]).

[itex]\psi ''(x)+\psi(x)=\phi ''(x)+\phi(x)=0[/itex]

Assume [itex]f(x)[/itex] be a continuous function defined in [itex]I[/itex], and let [itex]G(x,y), u(x)[/itex] as followings.

[tex]G(x,y)=\Bigg\{\begin{array}{l}\phi(x)\psi(y)\quad (0\leq y\leq x\leq \pi) \\ \psi(x)\phi(y)\quad (0\leq x\leq y\leq \pi) \end{array}

\\\\

u(x)=\int^\pi_0 G(x,y)f(y)dy[/tex]

Now my question is, what would be a proof for the equation: [itex]u''(x)+u(x)=Wf(x)[/itex] (for [itex]\exists W[/itex] is a constant).

I found u''(x)+u(x) constantly becomes 0 (for reason that [itex]u''(x)=\displaystyle{\int^\pi_0 \frac{\partial^2G(x,y)}{\partial{x}^2}f(y)dy} = -u(x)[/itex]).

Any hidden concept or my ignorance makes this so hard?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Need help solving functional analysis problem

**Physics Forums | Science Articles, Homework Help, Discussion**