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Need help solving a Non-Homogenous DE

  1. Apr 10, 2008 #1
    I need help solving a higher order differential equation by reduction of order.
    It will be greatly appreciated if all steps are posted as well!

    y(Double Prime)-3y(Prime)+2y=5e^3x where y sub one =e^x.
     
  2. jcsd
  3. Apr 11, 2008 #2

    lurflurf

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    Homework Helper

    "It will be greatly appreciated if all steps are posted as well! "
    um no, well maybe with the best bits obfuscated :)
    let D be differentiation w/ respect to x
    let y=u(x)*exp(x)
    L=(D^2-2D+2)
    Ly=5exp(3x)
    or
    (D^2-2D+2)y=5exp(3x)
    provided Lexp(x)=0 (is it? you should know)
    we will have transformed the equation (by substitution) into
    (aD^2+bD+c)u=f(x)
    where c=0 (show this is true in general)
    then
    u=[(aD^2+bD)^-1] f(x)
    or if you prefer find an integrating factor v(x) so that
    v(x)(aD^2+bD)u(x)=D(v(x)Du(x))
    thus
    (aD^2+bD)u(x)=f(x)
    becomes
    D(v(x)Du(x))=v(x)*f(x)
    and easy to solve ie
    u(x)=(1/D)(1/v(x))(1/D)[v(x)*f(x)]
    notice how each (1/D) yeilds one arbitrary constant as expected
     
    Last edited: Apr 11, 2008
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