Thread: Substitution
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Old April 30th, 2008, 01:36 PM
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Originally Posted by ubhik View Post
I have the following equation in my notes and need to follow it to do another question, can someone tell me how the following line:

(Ly)=(\frac{d^2}{dx^2}-\frac{2}{x}\frac{d}{dx}+\frac{2}{x^2})y(x)=xln(x)=h(x)

The kernel of L is found by solving Ly = 0

This form suggests trying y = x^m. Substituting,

(I don't understand how the following line is obtained??? )

L(x^m) = [m(m-1)-2m+2]x^{m-2}= 0

How does m turn into the base?

I know that:

y=x^m

\frac{dy}{dx}=mx^{m-1}

and

\frac{d^2y}{dx^2}=m(m-1)x^{m-2}

but I still don't understand, how the previous bit is obtained!
L(x^m) = \frac{d^2}{dx^2} \left ( x^m \right ) - \frac{2}{x} \frac{d}{dx} \left ( x^m \right ) + \frac{2}{x^2} \cdot x^m

= m(m - 1)x^{m - 2} - \frac{2}{x} \cdot mx^{m - 1} + \frac{2}{x^2} \cdot x^m

= m(m - 1)x^{m - 2} - 2mx^{m - 2} + 2x^{m - 2}

= [m(m - 1) - 2m + 2]x^{m - 2}

I'm not sure what else to give you, unless I'm completely misunderstanding your question.

-Dan
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