Solving differential equation using the Power Series Method

[jmg498]

Homework Statement

x²y'' + (x² - 2)y = 0

Homework Equations

N/A

The Attempt at a Solution

I divided through by x² to get the equation in standard form. Then I plugged in the power series representation for y and y'' into the equation and got to this point...

http://www.meteo.psu.edu/~jmg498/equ.png

Now, I know that I want the powers on x to be the same. But before I do an index shift, I'm not really sure how to handle the (x² - 2)/x² in front of the sum in the second term. If it were just an x, then I would be able to strip one element off the index. But since it's not just an x or power of x, I'm not really sure how to handle it.

Thanks for any hints!

 

[NoMoreExams]

Well

$$\frac{x^{2} - 2}{x^{2}} = 1 - \frac{2}{x^{2}}$$

And then just distribute the sum?

 

[jmg498]

That's where I'm confused. I'm not sure how to distribute the sum with that.

1 - 2/x² = 1 - 2x^-2 so does that mean the sum in the second term becomes x^n-2? Not sure...

 

[NoMoreExams]

Well you will now have 2 sums...

So if by the 2nd term you mean 2nd sum, then yes (with a 2 out front)