Solving PDE Homework Statement - Can You Help?

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Homework Statement



oh! after trying to re-solve a PDE I reached this:


Homework Equations


\sum\frac{4}{((2n-1)\pi)^2} (a+\frac{4(-1)^{n+1}}{(2n-1)\pi}) cos(\frac{2n-1}{2}\pi x)

n goes feom 1 to \infty and "a" is a constant value.

The Attempt at a Solution


the solution i am trying to reach is:

=\frac{1}{2} (1-x^{2}+(1-x)a)

but i don't know how?
can anyone help please?
 
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What is the Fourier series for \frac{1}{2} (1-x^{2}+(1-x)a)?
 
thanx for suggestion my buddy.
u know the orginal problem is a heat equation - one dimensional and time dependent-

T_{xx}+j^{2}=T_{t}
T_{t}=-1/2j\frac{b}{cL}
T(1,t)=0
T(x,0)=0

j,c,b are constant and 0\leqx\leq1

i solved the problem to here:

T(x,t)= j^{2} \sum (\frac{1}{\lambda_{n}^{2}}) \times \frac{b}{cL}+ \frac {2 \times -1^{n+1}}{\lambda_{n} cos(\lambda_{n}x + j^{2} \sum \frac{-1}{(\lambda_{n})^{2}}(e)^{-\lambda_{n}t} \left[\frac{b}{cL}+ \frac {2\times -1^{n+1}}{\lambda_{n}\right] cos(\lambda_{n}x)
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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