Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A differential equation question

  1. May 26, 2014 #1
    Verify that given function is a solution.
    y'' - 2y' + 2y = 0 , y=e^x(Acos x + Bsin x)

    First I take derivative of y which is y+e^x(-Asin x + B cos x) then I asign e^x(-Asin x + B cos x) to y'-y. Then I take derivative of y' which is y'+(y'-y)-y which equals 2y'-2y=y'' then I use y'' as 2y'-2y which satifies. Do you have a faster or more exciting way?

    Have a nice day.
     
  2. jcsd
  3. May 26, 2014 #2

    Bill_K

    User Avatar
    Science Advisor

    Just solve the DE. Let y = emx. Plug into the DE and get the condition m2 -2m + 2 = 0, which has solutions m = 1 ± i.

    Thus the DE has solutions y = a e(1 + i)x + b e(1 - i)x = ex(a eix + b e-ix), which by Euler's formula can be reduced to the form you give.
     
  4. May 26, 2014 #3

    lurflurf

    User Avatar
    Homework Helper

    That equation begs for the change of variable y=e^x u
    y'=e^x(u'+u)
    y''=e^x(u''+2u'+u)
    so
    y'' - 2y' + 2y = 0 , y=e^x(Acos x + Bsin x)
    becomes
    u''+u=0 , u=Acos x + Bsin x
     
  5. May 26, 2014 #4

    Bill_K

    User Avatar
    Science Advisor

    The method I gave works for any linear homogeneous DE with constant coefficients, reducing it to an algebraic equation. And in this case all that remains to do is solve a quadratic.

    Would you have been able to guess this change of variable if you did not already have the solution to look at?
     
  6. May 26, 2014 #5

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    The question doesn't ask the OP to solve the equation. It asks to verify the given function is a solution.

    Of course this is a very easy ODE to solve if you know how to solve it, but the question might not even be from a differential equations course. It might just be an exercise in differentiation.
     
  7. May 27, 2014 #6

    lurflurf

    User Avatar
    Homework Helper

    Not sure what you are getting at. Solving polynomials (other than quadratics) is not easy.
    Sure
    m2 -2m + 2=(m-1)2+1
     
    Last edited: May 27, 2014
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A differential equation question
Loading...