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Help solving a Cauchy-Euler Equation (Differential equation help)

  1. Feb 27, 2010 #1
    1. The problem statement, all variables and given/known data
    x2y'' + xy' + 4y = 0


    2. Relevant equations
    y = xr
    y' = r xr-1
    y'' = r(r-1)xr-2


    3. The attempt at a solution

    x2[r(r-1)xr-2] + x [r xr-1] + 4xr = 0

    (r2-r)xr + r xr + 4xr = 0

    [(r2-r) + r + 4]xr = 0

    r2 - r + r + 4 = 0

    r2 + 4 = 0

    from the quadratic equation i know that: r = 2i and r = -2i

    so y = c1x2i + c2x-2i

    my question is how can i remove the imaginary number from the answer.

    Thanks in advanced!
     
  2. jcsd
  3. Feb 27, 2010 #2

    ideasrule

    User Avatar
    Homework Helper

    x^2i can be expressed as e^(ln x^2i). Use Euler's equation to express that in terms of sine and cosine and you're done.
     
  4. Feb 27, 2010 #3
    If the roots of your characteristic equation [itex]r^2 + (b-1)r + c = 0[/itex] (here a = 1, where a is the coefficient of r^2, b = 1, and c = 4) are complex (occurs since b^2 - 4ac < 0), then the general solution is

    [tex]y= c_1 e^{\alpha x}\cos{\beta x} + c_2 e^{\alpha x}\sin{\beta x}[/tex]

    where

    [tex]r_1 = \overline{r_2} = \alpha + i\beta = \frac{-b}{2a} + i\frac{\sqrt{4ac - b^2}}{2a}[/tex]

    so that

    [tex]\alpha = \frac{-b}{2a} \mbox{ and } \beta = \frac{\sqrt{4ac - b^2}}{2a}[/tex]

    (All you are doing is reducing Cauchy-Euler equation to a homogeneous linear differential equation with constant coefficients)
     
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