While I was working on a variational problem, I met this equation as its Euler-Lagrange equation, but I cannot solve it:
$ x= \frac{af'(x)}{\sqrt{1+af'(x)^{2}}} + \frac{bf'(x)}{\sqrt{1+bf'(x)^{2}}} \ (a\neq b) $.



An ordinary differential equation
You can solve it parametrically as follows: Write 
$$
x(t) = \frac{at}{\sqrt{1+at^2}}+\frac{bt}{\sqrt{1+bt^2}}
$$
and
$$
y(t) = c - \frac{\sqrt{1+at^2}+\sqrt{1+bt^2}}{\sqrt{1+at^2}\sqrt{1+bt^2}}
$$
where $c$ is a constant. Then this gives the general solution as $c$ varies.


If you want an explicit relation between $x$ and $y$, you can eliminate $t$, but it won't be pretty.


