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This may be a silly question, but I am confused with the following. To my knowledge, in general any initial value problem we have a differential equation of the form $\dot{y}(x)=f(x,y(x))$ plus an initial condition, where $f$ is a real valued function. Now suppose that I have the following equation: $\dot{y}(x)=f(x,y(x),y)$, meaning that for any given pair $(x,y(x))$, $f(\cdot)$ becomes a functional of $y$. Here is my confusion...how should I label this problem? Is it still an initial value problem or is it a completely different animal? If so, what kind of animal is it?

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not sure i understand, could you give an example of your new formulation? –  gt6989b Sep 21 '12 at 19:41
@gt6989b for instance $\dot{\phi}(x)=x\phi(x)+\int_{0}^{x} H(\phi(s))ds$, where $H$ is some continuous function. –  Cristian Sep 21 '12 at 19:43
that would be an integro-differential equation (has both integrals & derivs), but differentiating both sides converts it to a pure differential equation of a higher order. –  gt6989b Sep 21 '12 at 19:44
@gt6989b so, with appropriate initial conditions, the problem can still be treated as an initial value problem? –  Cristian Sep 21 '12 at 19:47
I would expect so, yes. –  gt6989b Sep 21 '12 at 19:54

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