Solve and find at least 3 non-zero terms of each infinite series solutions to the differential equation about x=0 for 2x^2y" + xy' +x^2y = 0. Please show steps. Thanks!

Hello, please consider the following.

The equation is

Assume that, on a given domain where the sum is defined,

is a solution of the equation and we will find a recursion formula for the , then

So the equation becomes

And this is equal to 0, so we can say that

It comes

So

We can go further to find a generic expression but only 3 non-zero terms were requested.

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

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step-by-step explanation: