# maximal solution of differential or integral equation.

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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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.