# Series solution of first order ODE

1. Oct 22, 2009

### soverylost

1. The problem statement, all variables and given/known data

Find two non-zero terms of the power series solution of
y' = 1 + y^2 ,y(0) = 0
by using series substitution y(x) = sum (k=0 to inf) [a][/k] *x^k

2. Relevant equations

3. The attempt at a solution

First take the derivative of the power series to get
y' = sum (k=0 to inf) (k+1)*[a][/k+1]*x^k

Plug y and y' into the original ODE, here is where my problem is.
I want the powers of x to match so that i can match the coefficients of the series and get a recursive relationship to find the non-zero terms. How do i deal with the y^2 term? How do I square a series and still get matching x-terms?

2. Oct 22, 2009

### LCKurtz

Remember, you are only asked to find the first couple of terms. So you just multiply out the first few terms the long way. For example, to start multiplying out these two:

$$a_0 + a_1x + a_2x^2 + ...$$
$$b_0 +b_1x + b_2x^2 + ...$$

You would get:

$$a_0b_0 + (a_0b_1 + a_1b_0)x + ...$$

and do you see how to get all the x2 terms if you need them? So do that method for multiplying the series for y by itself for as many terms as you need.