solving second order ODE with $\operatorname{sech}^2(x)$ I am trying to solve the equation of the following type:
$$u''(x) = (a-b\operatorname{sech}^2(cx))u(x),$$
where $a,b,c$ are positive real constants.
It is my first time when I met the equation which includes $\operatorname{sech}^2(x)u$ function.
Any hint will be good for me! Thank you in advance!

There is a class of solution possible provided you allow for a relation between coefficients a and b. Try the solution $(\operatorname{sech}(cx))^p$ and find two algebraic equations detailing the dependence of the power $p$ on $a$ and $b$.
