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Homework Help: Help with Solving a Cauchy-Euler Differential Equation

  1. Mar 17, 2010 #1
    1. The problem statement, all variables and given/known data

    x2 y'' + x y' + 4 y = 0

    2. Relevant equations

    y = xr
    y' = r xr-1
    y'' = (r2-r)xr-2

    3. The attempt at a solution

    x2{(r2-r)xr-1} + x{r xr-1} + 4xr

    r2 - r + r + 4

    r2 + 4 = 0

    r = +- 2i

    y = C1x2i + c2x-2i

    My question is how can i remove the imaginary number with cos[] and sin[]
    If you could be as descriptive as possible i'd really appreciate it!

    Thanks in advanced!
  2. jcsd
  3. Mar 18, 2010 #2


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    Use the definition of exponentiation: xa=ea log x.
  4. Mar 18, 2010 #3


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    In fact, the substitution y= ln(x) will change any "Cauchy-Euler" equation into an equation with constant coefficients with the same characteristic equation. An equation with constant coefficients, with [itex]\pm 2[/itex] as characteristic roots, has general solution C cos(2y)+ D sin(2y).
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