Solving PDE Homework Statement - Can You Help?

[jahandideh]

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?

 

[HallsofIvy]

What is the Fourier series for $$\frac{1}{2} (1-x^{2}+(1-x)a)$$?

 

[jahandideh]

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$$\leq$$x$$\leq$$1

i solved the problem to here:

$$T(x,t)= j^{2} \sum (\frac{1}{\lambda_{n}^{2}}) \times \frac{b}{cL}+ \frac {2 \times -1^{n+1}}{\lambda_{n} cos(\lambda_{n}x + j^{2} \sum \frac{-1}{(\lambda_{n})^{2}}(e)^{-\lambda_{n}t} \left[\frac{b}{cL}+ \frac {2\times -1^{n+1}}{\lambda_{n}\right] cos(\lambda_{n}x)$$