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Consider the DE $$y''(x)+(1-x)^{-1}y'(x)+y(x)=0$$ where it has an ordinary point at x=0.

a) There are two different power series solutions about x=0; what is their radius of convergence?

b) Let these solutions be $$y=\sum_{n=0}^\infty a_nx^n$$ Find the corresponding recurrence solution.

c) Obtain the solutions up to and including the term in $$x^4$$ first for $$a_0 = 1, a_1 = 0, \textrm{ then for } a_0 = 0, a_1 = 1$$

So my understanding is that I am supposed to plug in the respective series for y, y', y'' into the original equation then get a recurrence relationship from it. But I am lost on the concept of radius of convergence and on how to start on c).

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hint

Write your equation as

$$(1-x)y''+y'×(1-x)y=0$$

with

$y=a_0+a_1x+a_2x^2+a_3x^3+$ $$...+a_nx^n+...$$

$y'=a_1+2a_2x+$ $$...+(n+1)a_{n+1}x^n+...$$

$y''=2a_2+2.3a_3x+$ $$...+(n+1)(n+2)x^n+...$$

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To find the radius of convergence, I would suggest using the Ratio Test. Remember that the Ratio Test asks you to calculate $$\displaystyle\lim_{n \to \infty} \left| \frac{a_{n + 1}}{a_n} \right|$$ You might get this value from the recurrence relation.

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