(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the indicial roots of the following Differential Equation: xy'' - y' + x^{3}y = 0

2. Relevant equations

y = Ʃ[n=0 to infinity]c_{n}x^{n+r}

y' = Ʃ[n=0 to infinity](n+r)c_{n}x^{n+r-1}

y'' = Ʃ[n=0 to infinity](r+r)(n+r-1)c_{n}x^{n+r-2}

3. The attempt at a solution

Plugging these values into the differential equation, I got

x^{r}{Ʃ[n=0 to infinity](n+r)(n+r-1)c_{n}x^{n-1}- Ʃ[n=0 to infinity](n+r)c_{n}x^{n-1}+ Ʃ[n=0 to infinity]4c_{n}x^{n+3}} = 0

The three sums must produce the x to the same exponent, so I tried pulling out the first 4 terms of the first two sums, so the three sums would each output x^{3}as their first term [the first two sums starting from n=4]. However, this left me with the following equation:

r(r-1)c_{0}x^{-1}- rc_{0}x^{-1}+ r(r+1)c_{1}- (r+1)c_{1}+ (r+1)(r+2)c_{2}x - (r+2)c_{2}x - (r+2)(r+3)c_{3}x^{2}- (r+3)c_{3}x^{2}+ [remaining sums] = 0.

How do I solve for r with this equation? I don't know how to find the roots.

[the solution to the DE is y=c_{1}cos(x^{2}) + c_{2}sin(x^{2})

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# Differential Equations, Frobenius' Method

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