Solving 2nd Order ODE: Even Function Solution

[omyojj]

Could you please help me or give me any hint to solve this ODE..

$$\frac{d^2y}{d x^2} + ( 2\rm{sech}^2 x - a^2 ) y = 0$$

where a is a constant.

I want only even function solution. (y(x) = y(-x))

BTW, this is a homework problem. I encountered this equation while considering surface waves in a self-gravitating incompressible fluid with stratification.

 

[jackmell]

You probably need to post this then in the homework forum where you are required to show some effort. Look, this is what I would do: I'd study it for a while, see if I can do something about it, I couldn't initially, then just resort in desperation to Mathematica. Now, if you don't have it, then you either need to find a machine running it or just plug it into Wolfram Alpha. Mathematica gives a solution and that solution is in terms of a particular well-known DE. At that point you can either just use it if you're an engineer or something, or if you like math, you might try and figure how to convert your DE into that particular DE for which the solution is given in terms of and therefore "figure" how to solve it. That is, show how your equation can be written as:

$$(1-u^2)y''-2uy'+\left(2-\frac{a^2}{1-u^2}\right)y=0$$

for $u=f(x)$ for appropriate $f(x)$.

 

[omyojj]

Oops, I committed an error...was going to say 'this is NOT a homework problem'..

$$x = \mathrm{tanh}u$$

and the resulting equation is the associated Legendre's equation.
Thank you anyway, I should've examined the equation with more patience..