Help witha second-order nonlinear ODE

Question

I'm stuck with the following second-order nonlinear differential equation: $$ \frac{Af'(x)\left(1+f'(x)^2\right)+(B+Ax)f''(x)}{\left(1+f'(x)^2\right)^{3/2}}=A+C $$ where $f(x)$ is the function that I want to find, while $A$,$B$, and $C$ are constant parameters. I've been searching for an exact solution but I'm having a hard time dealing with it. Do you think that there is an exact solution for it or should I go for a numerical method instead? Thank you in advance.

Answer 1

In such cases involving expressions $\sqrt{1+g(x)^2}$ you can try to set $g(x)=\sinh(u(x))$ or $g(x)=\frac12(v(x)-\frac1{v(x)})$. Using the first variant gives $$ A\tanh(u(x))+(B+Ax)\frac{u'}{\cosh^2(u(x))}=A+C $$ The left side is now recognizable as a product derivative, leading to $$ (Ax+B)\tanh(u(x))=(A+C)x+D. $$ Now solve this for $f'(x)=\sinh(u(x))$. It is unlikely that the resulting expression is symbolically integrable.