# Differential Eq- Power Series Solution

Find a power series sol'n: (x2-1)y'' + 3xy' + xy = 0

## Homework Equations

let y = $$\Sigma$$ (from $$\infty$$ to n=0) Cnxn
let y' = $$\Sigma$$ (from $$\infty$$ to n=1) nCnxn-1
let y'' = $$\Sigma$$ (from $$\infty$$ to n=2) n(n-1)Cnxn-2

## The Attempt at a Solution

I wrote the differential eq as: x2y''-y''+3xy'+xy=0

Substituting back into the differential eq and multiplying the x2 gives:
$$\Sigma$$ (from $$\infty$$ to n=2) n(n-1)Cnxn - $$\Sigma$$ (from $$\infty$$ to n=2) n(n-1)Cnxn-2 + 3$$\Sigma$$ (from $$\infty$$ to n=1) nCnxn + $$\Sigma$$ (from $$\infty$$ to n=0) Cnxn+1 = 0

For the first 2 terms I let k=n-2 , n=k+2 which would give:
$$\Sigma$$ (from $$\infty$$ to k=0) (k+2)(k+1)Ck+2xk+2 - $$\Sigma$$ (from $$\infty$$ to k=0) (k+2)(k+1)Ck+2xk

For the 3rd and 4th term I let k=n which would give:
3$$\Sigma$$ (from $$\infty$$ to k=1) kCkxk + $$\Sigma$$ (from $$\infty$$ to k=0) Ckxk+1

The new series would be:
$$\Sigma$$ (from $$\infty$$ to k=0) (k+2)(k+1)Ck+2xk+2 - $$\Sigma$$ (from $$\infty$$ to k=0) (k+2)(k+1)Ck+2xk + 3$$\Sigma$$ (from $$\infty$$ to k=1) kCkxk + $$\Sigma$$ (from $$\infty$$ to k=0) Ckxk+1

I don't know how to make the starting value of each index the same (from $$\infty$$ to k=0) and how would I get to the general solution?

## Answers and Replies

Cyosis
Homework Helper
First off a tip. Write your summations like this: \sum_{n=a}^\infty. This will look like $\sum_{n=a}^\infty$, which is a lot neater. Secondly put the tex brackets around your entire equation, not just the summation symbols.

This will make your equation look like:
$$\sum_{n=0}^\infty c_n x^n$$

Let's start with a simple example. Take $\sum_{n=1}^\infty c_{n-1}x^{n-1}$. We want to write this series as $\sum_{n=0}^\infty c_k x^k$. Lets take a look at the first series.

$$\sum_{n=1}^\infty c_{n-1}x^{n-1}=c_0 x^0+c_1 x^1+c_2 x^2+...$$

We obviously want $\sum_{n=0}^\infty c_k x^k$ to be equal to this. So how would you express k in terms of n?

$$\sum_{n=0}^\infty c_k x^k=c_0 x^0+c_1 x^1+c_2 x^2+...$$?

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