Solve for $f(x)$ in $b( a+f(x) )f'(x)f'(x) = c$

Question

I don't know how to solve the nonlinear differentiable equation $b( a+f(x) ) f'(x) f'(x) = c$ where $a,b,c$ are just constants and $f: [0,\infty] \to [0,\infty]$.

Answer 1

If you let $y=f(x),$ and then $z=a+y,$ then $z'=y',$ and the DE simplifies to \begin{align*} z(z')^2&=c/b\\ \sqrt{z}\,z'&=\pm\sqrt{c/b}:=k\\ \int z^{1/2}\,dz&=k\int dt\\ \frac{z^{3/2}}{3/2}&=kt+C\\ z^{3/2}&=\frac{3kt}{2}+C\quad\text{absorb 3/2 into }C\\ z&=\left(\frac{3kt}{2}+C\right)^{\!2/3}\\ y&=\left(\frac{3kt}{2}+C\right)^{\!2/3}-a\\ &=\left(\pm\frac{3\sqrt{c}\,t}{2\sqrt{b}}+C\right)^{\!2/3}-a. \end{align*} You can plug this into the original DE and show that it works. Also, depending on the values of $a,b,c,$ you may be able to eliminate one of the $\pm,$ since you are requiring a non-negative function.