Second Order Linear Differential Equation

[Pawnag3]

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!

 

[Mugged]

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?

 

[Pawnag3]

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.

 

[Mark44]

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]

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.

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!

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

 

[Ray Vickson]

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!

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?