I was wondering if there was any particular reason that power series are centred around 0.
$$\displaystyle f(x)=\sum_{k=0}^{\infty}c_{k}x^{k}\qquad \text{ vs }\qquad\displaystyle f(x)=\sum_{k=0}^{\infty}c_{k}(x-a)^{k}$$
when using them to solve differential equations. I've tried looking online for a reason but so far have had no luck.
This seems like a rather arbitrary choice that only conveniently simplifies our computation – eg. what if the function is not well-defined at 0 or deviates significantly away from 0?
Apologies if this seems like a trivial question – I have not done a differential equations course yet that involves power series. Solutions involving only first-year university maths would be much appreciated.
Best Answer
Imagine you have the following differential equation: $$\frac{d^2y}{dx^2}+\frac{dy}{dx}+x+c=0 \tag{*}$$ Subjected to some boundary conditions.
You can say that your solution $y$ may be expanded as a polynomial series: $$y=\sum_k^N c_k(x-x_0)^k$$ With $c_k$ and $x_0$ constants.
since the solution of the problem $(*)$ is unique, the polynomial $(**)$ is also unique, and you can solve for $c_k$ or for other coefficients: $$y=\sum_k^N c_k(x-x_0)^k=\sum_k^N \hat{c}_kx^k$$ where there is a correspondence between coefficients: $$c_N=\hat{c}_N, \quad c_{N-1}=c_{N-1}-c_Nx_0,...$$