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My question is rather silly. What is the definition of maximal solution to an integral equation and differential equation. For example, how should I understand the maximal solution of the following $\frac{d u}{d x}=\sqrt{u}$?

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As stated here A maximal solution to a differential equation x'= f(x) is a solution defined on an interval I such that there is no solution defined on an interval I', which properly contains I.

In your problem, by separation of variables you get: $$\frac{du}{\sqrt{u}}=dx$$ and by integrating:$$2\sqrt{u}=x+c$$ since you're looking for a maximal solution, the biggest domain that this function is defined would be $[-c,+\infty)$ and the solution of your ODE would be $$u(x)=\frac{(x+c)^2}{4}, \ x\in [-c,+\infty)$$c can be calculated by using a boundary condition.

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In general, maximal solution refers to the domain of existence of the solution. You are looking for the solution of your equation that has the biggest possible domain of definition. For differential equations (at least for those that have some reasonable regular behavior), maximal solutions exist if you have a theorem about local solvability and uniqueness.

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