I have a recursive integral,
$f(T) = f(0) + \frac{\gamma}{2} \int_{0}^T f(t)^2 dt$
where $f(0) \in \mathbb{R}^+$, $\gamma \in \mathbb{R}$ and $T \in \mathbb{R}^+$ all take known numerical values.
Is there some way to convert the above equation for $f(T)$ to closed form? I'm completely stumped. It resembles an equation of motion in some sense but that's not helped me.
Thanks for any thoughts!
Differentiating the equation gives $$ f'(T)=\frac{\gamma}{2}f^2(T). $$ So one has $$ \frac{f'(T)}{f^2(T)}=\frac{\gamma}{2} $$ which implies $$ \int\frac{df(T)}{f^2(T)}=\frac{\gamma}{2}T+C $$ or $$ -\frac{1}{f(T)} = \frac{\gamma}{2}T+C. $$ Thus $$ f(T)=-\frac{1}{\frac{\gamma}{2}T+C}. $$ But $f(0)=-\frac{1}{C}$, one has $C=-\frac{1}{f(0)}$ and hence $$ f(T)=-\frac{1}{\frac{\gamma}{2}T-\frac{1}{f(0)}}=-\frac{f(0)}{\frac{\gamma}{2}f(0)T-1}. $$