Non-linear non-homogeneous first order ODE

Question

I'm trying to find the solution to the following ODE:

$$ (y'(x))^2 - \left(\frac{y(x)}{\epsilon}\right)^2 + f(x)^2 = 0, $$ where $f(x)$ is some given function and I want to solve for $y(x)$. Any help would be appreciated.

Answer 1

Write the differential equation as $$y'(x) = \pm \sqrt{y(x)^2/\epsilon^2 - f(x)^2}$$

To speak of "the solution" rather than "a solution", you'll need to specify an initial condition $y(x_0) = y_0$. I'm assuming you want real solutions, so $y_0^2/\epsilon^2 - f(x_0)^2$ had better not be negative. Also you should specify whether you want $+$ or $-$ (a solution can switch from one to the other, but only when the square root is $0$).

There's no hope of a closed-form solution with $f$ left as an arbitrary function. Even for very simple $f$ such as $f(x)=x$, I don't think there are closed-form solutions. But for particular $f$ and particular initial condition you'll be able to find numerical solutions. You should be aware that the solutions are likely to stop existing $y^2/\epsilon^2 - f(x)^2$ hits $0$ and threatens to become negative.