Hi everyone,

I faced a problem in solve a second order ODE by power series. Hope someone help me out there:

Here is the question:

This is my solution (I have checked many times, probably nothing wrong):

Let

Then,

After shift the index,

Therefore a recurrence relation, for all n > 0:

To find terms in only & ,

For n = 0,

For n = 1,

For n = 2,

For n = 3,

.....

Since

As far as I know,

In fact, when I solve using charactheristic equation,

And the solution is

Now, here my question:

Is ?

If yes, please can you help me proof it?

If not, could you point out where my mistake?