Liner differentials of order n, Kernel

[bakin]

Homework Statement

Verify that the given function is in the kernel of L.

y(x)=x^-2
L = x²D² + 2xD - 2

Homework Equations

The Attempt at a Solution

I took the first and 2nd derivative of y(x), and got
y^'(x)= -2x^-3
y^''(x)= 6x^-4

Then plugged it into L (and a little simplifying) and got

L(y) = 6x^-2+2x^-1-2

I think I'm supposed to plug it in, and verify that it's equal to zero, but it's not coming out right.

Any obvious mistakes? Or wrong direction all together?

 

[Dick]

How does 2xDy become 2x^(-1)?? And the -2 isn't just a -2. L is operating on y. What should it be?

 

[bakin]

Bah, forgot about the -2 part. It's actually -2y, correct? So the last term would be -2y, or -2x^-2.

And as I was typing out how I came up with 2xD, I realized I substituted just y into D, and not y^'

With the correct substitutions, I came up with:

L = x²*6x^-2 + 2x*-2x^-3 - 2x^-2

= 6x^-2 - 4x^-2 - 2x^-2

= 0


Once again, thanks Dick.