I have to solve the nonlinear first-order differential equation $$\frac{a-y'}{\sqrt{1+y'^2}}e^{-a \arctan y'}=bx+c,$$ where $a,b,c$ are constants, and $y$ is a function of $x$.
Obviously, there is no way to put it in the form $y'=f(x)$ and integrate. However, I know that a simple parametric solution exists: $$\begin{align} x(t) &= c_1 + c_2 e^{-a t}(a\cos t-\sin t) \\ y(t) &= c_3 + c_2 e^{-a t}(a\sin t+\cos t), \end{align} $$ for some constants $c_1,c_2,c_3$ (that depends on $a,b,c$). How can we arrive at this solution? What is the method?