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Answer check: differential initial value problem

  1. Apr 26, 2009 #1
    1. The problem statement, all variables and given/known data
    solve the IVP

    y''-8y'+16y=0

    y(0)=2
    y'(0)=7


    2. Relevant equations



    3. The attempt at a solution

    auxiliary equation:
    r2-8r+16=0
    (r-4)2=0
    so we have r=4 with m=2

    y(x)= c1e4x+c2xe4x

    y(0)--> c1e0 + 0 = 2
    c1=2
    now we have:
    y(x)=2e4x+c2xe4x

    take first derivative
    y'(x) = 8e4x+c2e4x+4c2xe4x (product rule)

    y'(0) ----> 8e0+c2e0+ 0 = 7
    8+c2=7
    c2=-1

    so our final equation is:

    y(x)=2e4x-xe4x

    Everything seems ok, but just wanted to run it by here to make sure it's all done correctly. thanks all!
     
  2. jcsd
  3. Apr 26, 2009 #2
    Looks fine to me
     
  4. Apr 27, 2009 #3

    Mark44

    Staff: Mentor

    It's much easier to check that a potential solution actually works than it is to get the solution. For your problem all you need to do are the following:
    1. Verify that y(x) = 2e4x + xe4x satisfies y'' - 8y' + 16y = 0.
    2. Verify that y(0) = 2.
    3. Verify that y'(0) = 7.

    If your solution satisfies the differential equation and initial conditions, you can bask in the warm glow of confidence that you nailed that problem.
     
  5. Apr 27, 2009 #4
    ......I didn't even think of trying that. Thanks guys!
     
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