Solving ODE: Get Help with y''(x)+(μ^2*c(x)+k^2)y=0

[gleonard]

I have no idea how to start solving this ODE:

y''(x)+(μ^{2}*c(x)+k^{2})y=0

Where c(x)=A+Btanh(ρx) with constant A,B,ρ,μ,k

Could anyone give me a nudge in the right direction?

Cheers.

 

[bigfooted]

There is not a general method for finding the solution (if it exists) to y'' + f(x)*y = 0, so if f(x) is complicated you usually need complicated algorithms and the solution cannot be found by hand anymore.

When the constants remain unspecified, the solution is a page long hypergeometric solution (computed with Maple). For certain specific values for the constants this could perhaps be reduced to a more compact solution, but the general solution is not simple.

 

[gleonard]

There is not a general method for finding the solution (if it exists) to y'' + f(x)*y = 0, so if f(x) is complicated you usually need complicated algorithms and the solution cannot be found by hand anymore.

When the constants remain unspecified, the solution is a page long hypergeometric solution (computed with Maple). For certain specific values for the constants this could perhaps be reduced to a more compact solution, but the general solution is not simple.

Thanks for your reply bigfooted,

I have been told I need to use hypergeometric solutions, (sorry for not mentioning this), but I have never seen them before. I have found the basic form of a hypergeometric series but I don't know how to apply this.

 

[Apophenia]

Do you require the solution to be analytic?

Edit: scratch that. I think you answered that in your reply

 

[Mute]

Thanks for your reply bigfooted,

I have been told I need to use hypergeometric solutions, (sorry for not mentioning this), but I have never seen them before. I have found the basic form of a hypergeometric series but I don't know how to apply this.

If you have to deal with hypergeometric functions, it may be better just to solve your equation numerically. There are some computer packages that calculate hypergeometic functions, but if you end up having to deal with something more complicated than the standard hypergeometric function $_2 F_1(a,b;c;z)$, it's probably not worth it to try and use them. I guess it depends on what you want to do with the solution. Are you interested in just what the solution looks like for certain parameter choices, or do you want to know how it changes with your parameters (e.g., the asymptotic properties of the solution)?

Your equation can be simplified by writing it in a non-dimensionalized form,
$$y''(t) + (\tilde{A} + \tilde{B}\tanh(t))y(t) = 0,$$
by making the change of variables $t = \rho x$ and writing $\tilde{A} = (\mu^2 A + k^2)/\rho^2$ and $\tilde{B} = \mu^2 B/\rho^2$. This is simpler for programs like Mathematica to handle.

Using Mathematica, for arbitrary $\tilde{A}$ and $\tilde{B}$, shows that the solution is indeed in terms of $_2 F_1$, so you could write down the closed form solution and use a routine for computing $_2F_1$, but it might not give you much advantage over just solving the equation numerically. You may be able to extract some asymptotic properties from the analytic solution, however.

If your goal is to demonstrate the solution by hand yourself, then you should probably try to make a change of variables to help cast the non-dimensionalized equation into the form of the hypergeometric equation,

$$z(z-1)\frac{d^2 w}{dz^2}+[c-(a+b+1)z]\frac{dw}{dz} - a b w = 0.$$

See wikipedia article for the set of possible solutions in terms of $_2 F_1$ for this equation.