Should I use the indefinite or the definite integral when working with differential equations?

Question

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?

Answer 1

A rule of thumb would be to use indefinite integrals when there is just a differential equation, and use definite integrals when it's an initial value problem. If the initial condition is $y(x_0)=y_0$, then the lower limit on the definite integral should be $x_0$.

But the rule of thumb isn't hard and fast. Sometimes, even without an initial condition, it's convenient to have the arbitrary constant explicit. The indefinite integral $\int f(x) \; dx$ has an implicit, but invisible, arbitrary constant. Sometimes, to do a derivation, one needs that constant explicitly written in the equation. Either one writes $\int f(x) \; dx+c$ (which is a bit reduntant, but correct) or fixes it with a definite integral $_{x_0}^x \int f(t) \; dt +c.$

Mostly, try to be clear for your reader.