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)
 
Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'
Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...
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