I've been working on a line integral and calculus of variations problem where I am trying to minimize the line integral for a given scalar function $f(x,y)$. I have these two coupled, nonlinear, first-order differential equations:
$f(x,y)\sqrt{\dot{x}^2+\dot{y}^2}-\frac{\dot{x}^2f(x,y)}{\sqrt{\dot{x}^2+\dot{y}^2}}=k$
$f(x,y)\sqrt{\dot{x}^2+\dot{y}^2}-\frac{\dot{y}^2f(x,y)}{\sqrt{\dot{x}^2+\dot{y}^2}}=l$
I figured out how to solve for the case when $f(x,y)=\sqrt{x^2+y^2}$. The solution was $x(t)=\cosh(t)$ and $y(t)=\sinh(t)$. I did this by subtracting the two equations from each other and turning the resulting fraction into two equations, where the first is the hyperbolic equation. However, I want to be able to solve these equations for other $f(x,y)$, such as $f(x,y)=x^2y^2$. I have not taken an ODE class. What techniques can I use to solve these equations other than the one I stated (which only works for that specific scalar function)?