Differential equation with power series method

[Schfra]

Homework Statement

I need to solve the DE

y’ = x^2y

using the power series method

Homework Equations

y = sum(0->inf)Cnx^n
y’ = sum(1->inf)nCnx^(n-1)

The Attempt at a Solution

I plug in the previous two equations into the DE. What is the general procedure for these problems after that?

I believe I can bring the x^2 into the summation to get

sum(1->inf)nCnx^(n-1) - sum(0->inf)Cnx^(n+2) = 0

I’m not sure what to do after this.

 

[tnich]

Now you need to solve for the coefficient of $x^n$ for each value of $n$. So for example $2c_2 - c_{-1} = 0$. It looks to me like you need to extend your series to include values of $n$ less than 0. Otherwise, all of your coefficients will be zero.

 

[Schfra]

It looks to me like you need to extend your series to include values of $n$ less than 0. Otherwise, all of your coefficients will be zero.

Can you explain what you mean by this and why I have to do this? I did get 0 for my coefficients while attempting to solve earlier.

 

[tnich]

Can you explain what you mean by this and why I have to do this? I did get 0 for my coefficients while attempting to solve earlier.

There are a couple of tricks here. First, assume that $nc_n - c_{n-3} = 0$ for all $n \in \mathbb Z$. You end up with a recursion. Actually you end of with three recursions. For two of them you easily compute a limit and dispense with them. The other one gives you $0c_0 - c_{-3} = 0$. What does that imply $c_{-3n}$? About $c_0$?

 

[Orodruin]

Now you need to solve for the coefficient of $x^n$ for each value of $n$. So for example $2c_2 - c_{-1} = 0$. It looks to me like you need to extend your series to include values of $n$ less than 0. Otherwise, all of your coefficients will be zero.

I disagree. The solution to the differential equation has a smooth behaviour around $x = 0$ so expanding it in a Maclaurin series should be perfectly fine.

Also note that you have no $c_{-1}$ so clearly $c_2 = 0$ (which agrees with the exact solution).