Differential Equations question

[jawhnay]

Homework Statement

Determine for which values of m the function ∅(x) = x^m is a solution to the given equation
a) 3x²y" + 11xy' -3y = 0
b) x² y" - xy' - 5y = 0

The Attempt at a Solution

I tried approaching this problem by substituting ∅(x) into the question.
a) 3x²(x^m)'' + 11x(x^m)' - 3(x^m) = 0
x^m(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.

 

[alanlu]

Check your differentiation. What is d²/dx²(x^m)?

 

[jawhnay]

y'= mx^m-1 y''=(m-1)(m)x^m-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?

 

[alanlu]

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

 

[jawhnay]

3x²(mx^m-1) + 11x(m-1)(m)x^m-2 - 3(x^m) = 0
3x²mx^m-1 + 11x(m²-m)x^m-2 - 3x^m = 0
x^m(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...

 

[alanlu]

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

 

[jawhnay]

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

 

[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!