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Solving second order linear homogeneous differential equation

  1. May 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the set of functions from [itex](-1,1)→ℝ[/itex] which are solutions of:

    [itex](x^{2}-1)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-4y = 0[/itex]

    2. Relevant equations

    3. The attempt at a solution

    There is a hint which says to use the change of variable:
    [itex]x=cos(θ)[/itex]

    doing this I get:

    [itex]\frac{dx}{dθ} = -sin(θ)[/itex]

    [itex]\Rightarrow[/itex]

    [itex]dx = -sin(θ)dθ[/itex]

    [itex]\Rightarrow[/itex]

    [itex](a): \frac{dy}{dx} = \frac{dy}{dθ}\frac{1}{-sin(θ)}[/itex]

    [itex](b): \frac{d^{2}y}{dx^{2}} = \frac{d^{2}y}{dθ^{2}}\frac{1}{sin^{2}(θ)}[/itex]

    Can I do this?!?

    If so, substituting everything in gives:

    [itex]-sin^{2}(θ)\frac{d^{2}y}{dθ^{2}}\frac{1}{sin^{2}(θ)} + cos(θ)\frac{dy}{dθ}\frac{1}{-sin(θ)} - 4y = 0[/itex]

    [itex]\Rightarrow[/itex]

    [itex]y'' + cot(θ)y' + 4y = 0[/itex]

    Now.... I am not sure.
    Have I made some mistake?
    Or should I be able to solve this?
    Could someone please point me in the right direction?
    Thanks a lot!
     
  2. jcsd
  3. May 9, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi the0! :smile:
    no, your (b) doesn't include dy/dθ d/dx(-1/sinθ)
     
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