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If given initial conditions for a separable differential equation, finding them is easy enough, for example if $y(0) = 0$

But how do I know that these are the only solutions? Or that this solution even exists at all before I start calculating it?

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Let your equation be $$f(y)\ dy=g(x)\ dx.$$

Taking the definite integrals from the initial point to the current one, you have$$F(y)-F(y_0)=G(x)-G(x_0).$$ This identity leaves no room for other solutions.

If $y_0$ and $x_0$ belong to the domain of $F$ and $G$, there is at least one solution point. So the question of existence relates to the integrability of $f$ and $g$ around the initial point.

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