How to solve a complicated ODE

Question

The equation is $$ f^{\prime}(x) = \gamma \frac{f(x)+f^2(x)}{\log\left(\frac{f(x)}{1+f(x)}\right)} $$ with the initial condition $f(0)$, where $x\geq 0$ and $f(0)\geq0$.

The solution is $$f(x)=\frac{1}{-1+\exp\sqrt{2\gamma x + \log\left(\frac{f(0)+1}{f(0)}\right)^2}}$$

I think the ODE can be solved by separating the variables $$ \int_{0}^{f(x)} \frac{\log\left(\frac{f(z)}{1+f(z)}\right)}{f(z)+f^2(z)} df = \int_{0}^{x} \gamma dz $$ The right-hand size is easy. I do not know how to solve the integration of the left-hand side.

Answer 1

Hint: Your method so far is correct. What you are asking for is the integral $$\int \frac{\log\left(\frac{y}{1+y}\right)}{y+y^2}~dy.$$ The integral at first looks deceptively hard to solve but it is actually very easy if you notice it is of the form $$\int g(y)g'(y)~dy$$ for $g(y)=\log\left(\frac{y}{1+y}\right)$.

Answer 2

Your approach is correct. As regards the integral on the left, note that $$\int \frac{1}{f(z)+f^2(z)} df = \int \left(\frac{1}{f(z)}-\frac{1}{1+f(z)}\right) df =\log\left(\frac{f(z)}{1+f(z)}\right)+c.$$ Can you take it from here?