I'm working through Pollard & Tenenbaum's "Ordinary Differential Equations".
In their treatment of first-order differential equations they write that for an equation of form
$$P(x)dx + Q(y)dy = 0$$
its solution is given by
$$\int P(x)dx + \int Q(y)dy = c $$
In other words, they take indefinite integrals to obtain the solution.
Later in the part on first-order equations, they arrive at exact differential equations, and use definite integrals in their proof of the exact differential equation's solution, to arrive at
$$\int_{x_0}^x P(x, y)dx + \int_{y_0}^{y} Q(x_0, y)dy = c$$
as a solution of
$$P(x, y)dx + Q(x, y)dy = 0$$
Where $P$ and $Q$ are partial derivatives of a multivariable function $f$.
When do I use the definite integral $\int_{x_0}^x f(x)dx$ and when can I use $\int f(x)dx$, taking just the antiderivative and being done with it?