<s>[INST] Finding Coefficients of Power Series Expansion Find the coefficients of $ a_{n}$ and $b_{n}$ for $ 0 ≤ n ≤ 4$ for the power series expansion of two linearly independent solutions of the ODE:  $y'' -(e^{x}-1)y=0$.
This is what I've tried so far:
$\sum_{n=2}^{\infty} n(n-1)c_{n}x^{n-2} - \sum_{n=0}^{\infty}\frac{1}{n!}x^{n}\sum_{n=0}^{\infty}c_{n}x^{n} + \sum_{n=0}^{\infty}c_{n}x^{n} = 0$
$\sum_{n=0}^{\infty} (n+2)(n+1)c_{n+2}x^{n} - \sum_{n=0}^{\infty}\frac{1}{n!}x^{n}\sum_{n=0}^{\infty}c_{n}x^{n} + \sum_{n=0}^{\infty}c_{n}x^{n} = 0$
Don't know how to continue from here.
 [/INST] Hint: $e^x = 1 + x + \dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\ldots$, $y_1(x) = x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \ldots$, and $y_2(x) = 1 + b_2 x^2 + b_3 x^3 + b_4 x^4 + \ldots$.
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