Differential Equations question

  • Thread starter jawhnay
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  • #1
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Homework Statement


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


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.
 

Answers and Replies

  • #2
65
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Check your differentiation. What is d2/dx2(xm)?
 
  • #3
37
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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
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Yes. Substitute them back in, but mind how the powers of x add up. What do you get?
 
  • #5
37
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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
65
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Shouldn't it be 3x2((m-1)(m)xm-2) + 11x(mxm-1) - 3(xm) = 0?
 
  • #7
37
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oh my god... I can't believe I didn't catch that. Okay, let me try to do this again.
 
  • #8
37
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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!
 

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