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Second Order Linear Differential Equation

  1. Mar 21, 2014 #1
    Hey, I'm not sure how to even approach this problem. It's not a simple ODE.

    Basically, I want to find the solution for Θ in terms of ε. The equation is
    [itex]\frac{1}{ε}*\frac{d}{dε}*(ε*\frac{dΘ}{dε})-β^{2}Θ=0[/itex]

    I tried to move the B^2 to the other side and I wasn't able to solve it that way. I can't solve it like a normal second order ODE because it has ε in front.

    Thanks for your help in advance!
     
  2. jcsd
  3. Mar 21, 2014 #2
    Can you define the notation a bit? Are you solving for Θ? Is Θ a function of ε -> Θ(ε)? Is β just a constant? are there any initial conditions or you just need a general solution?
     
  4. Mar 21, 2014 #3
    Sorry! I want to solve for Θ, and Θ is a function of ε, β is just a constant. I just want a general solution please.
     
  5. Mar 22, 2014 #4

    Mark44

    Staff: Mentor

    Part of your notation makes no sense. The equation above should not have d/dε "times" something. It means to take the derivative with respect to ε of (ε dθ/dε). You'll need to use the product rule to simplify this part.

    Once you do this, you'll have a second order DE to solve.
    On a side note, it would be much simpler to write the equation in terms of the letters that are usually used, rather than Greek letters. Translated to x and y, your equation looks like this:
    1/x * d/dx(x dy/dx) - β2y = 0
     
  6. Mar 22, 2014 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper


    What is stopping you from multiplying through by ε, so it will not have ε "in front"? Of course, you need ε ≠ 0, but you had to have that anyway, since you were initially dividing by ε.

    BTW: do you know about Bessel functions and Bessel's differential equation?
     
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