# differential equation with integral function

Could you please help me with any information which would allow me to get an explicit solution for this equation?

It is an implicit solution to an optimization problem involving resource allocation in a game-theoretic setting. (I do not think this type of problems have been considered.)

$$y'(x)= 1-\frac{2\sqrt{2 y(x)}}{\int_{0}^{x} \frac{dt}{\sqrt{y(t)+y(x)}}}$$

Which class of integro-differential equations would it belong - so that I could search for a solution?

It is a specific parametric example for a more general problem $$(2 y(x))^a f(x) = (1-y'(x)) \int_{x_{min}}^{x} a (y(t)+y(x))^{a-1} d F(t)$$ (given $a$ and distribution $F(x), f(x)=F'(x)$).

For $a=1$, the explicit solution is strikingly elegant: $$y(x) = \int_{x_{min}}^x \left(\frac{F(t)}{F(x)}\right)^2 dt$$

Many thanks in advance

• OK, seems to exist in English as well: Integro-differential equation – mvw Aug 15 '18 at 17:18
• Looks weird. Where did this show up? – mvw Aug 15 '18 at 17:19
• Thank you! Will look into the integro-differential equations.Though I am afraid mine might be a bit more complicated. (It comes as a first order condition to an unusual resource allocation problem with game-theoretic flavour.) – user584495 Aug 15 '18 at 17:48
• You will probably have to resort to some numeric method. – mvw Aug 15 '18 at 17:58
• Many thanks for the leads! – user584495 Aug 16 '18 at 14:11