# Liner differentials of order n, Kernel

## Homework Statement

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

y(x)=x-2
L = x2D2 + 2xD - 2

## 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?

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

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

:tongue: Once again, thanks Dick.