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Homework Help: Characteristic equation with x^2 coefficient

  1. Nov 27, 2017 #1
    1. The problem statement, all variables and given/known data

    x2 d2y/dx2 + 3x dy/dx + 5y = g(x)

    2. Relevant equations
    How do we find Characteristic equation for it.

    3. The attempt at a solution

    x2λ2 + 3xλ + 5 = 0
    λ1 = 1/2 [-x2 + √ (x4 + 20 ) ]
    λ2 = 1/2[ -x2 - √(x4 + 20) ]

    I used 1/3 -/+ a √(a2 + 4b)
    a = x2
    b = 5
  2. jcsd
  3. Nov 27, 2017 #2


    Staff: Mentor

    If memory serves, the characteristic equation applies to the related nonhomogeneous DE -- ##x^2 y'' + 3x y' + 5y = 0##.
    This is an example of an 2nd order Euler equation. One technique is to assume a solution of the form ##y = x^n##, and substitute it into the DE. For your nonhomogeneous equation, it depends on what g(x) is.
    See https://www.math24.net/second-order-euler-equation/
    I have no idea what you're doing here.
  4. Nov 27, 2017 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Forget characteristic equations: they do not apply in this problem. They are for constant coefficient linear DEs, but your DE has coefficients that are functions of ##x##.
  5. Nov 27, 2017 #4


    Staff: Mentor

    The old saying applies here: "If the only tool you have is a hammer, everything looks like a nail."
    As Ray said (and I forgot), characteristic equations are applicable only to constant coefficient linear differential equations, and specifically to homogeneous DEs of that type.
  6. Nov 28, 2017 #5
    If you need a quick reference, the method is called cauchy-euler (I think this is the name). There is also an equivalent form using a log, worth it to know both versions. t I believe you can also use Frobineous Method, due to the analytical point. Its been years since I solved a differential equation.
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