# Homework Help: Differential Equations question

1. Feb 25, 2012

### jawhnay

1. The problem statement, all variables and given/known data
Determine for which values of m the function ∅(x) = xm is a solution to the given equation
a) 3x2y" + 11xy' -3y = 0
b) x2 y" - xy' - 5y = 0

3. The attempt at a solution
I tried approaching this problem by substituting ∅(x) into the question.
a) 3x2(xm)'' + 11x(xm)' - 3(xm) = 0
xm(3x2m2 + 11xm - 3) = 0
I don't know what to do after this step and I'm not sure this is the correct way to do this problem. I just tried doing this because I looked at a similar question and this is how it was approached in the solutions manual.

2. Feb 25, 2012

### alanlu

Check your differentiation. What is d2/dx2(xm)?

3. Feb 25, 2012

### jawhnay

y'= mxm-1 y''=(m-1)(m)xm-2 Are you asking me to substitute these into the given equation? I actually did that but what am I going to do with the x variable since i'm looking for what m is equal to?

4. Feb 25, 2012

### alanlu

Yes. Substitute them back in, but mind how the powers of x add up. What do you get?

5. Feb 25, 2012

### jawhnay

3x2(mxm-1) + 11x(m-1)(m)xm-2 - 3(xm) = 0
3x2mxm-1 + 11x(m2-m)xm-2 - 3xm = 0
xm(3x2mx-1 + 11x(m2-m)x-2 - 3) = 0
m3x + 11x - 1(m2-m) - 3 = 0
m3x + m211x-1 - m11x-1 - 3 = 0
m(3x + m(11/x) - 11/x) = 3

I stopped right there since I wasn't really sure how I was going to get rid of the x...

6. Feb 25, 2012

### alanlu

Shouldn't it be 3x2((m-1)(m)xm-2) + 11x(mxm-1) - 3(xm) = 0?

7. Feb 25, 2012

### jawhnay

oh my god... I can't believe I didn't catch that. Okay, let me try to do this again.

8. Feb 25, 2012

### jawhnay

I finally got the answer. I can't believe one stupid mistake like that got me stuck on this problem. Thanks a lot for pointing out that mistake for me, alan!