Problem

Solve the following differential equation:

Sketch the family of solution curves.

Solution

First we rearrange the terms in the differential equation to separate the variables.

The left side is of the form 1/u, so we guess its antiderivative to be ln(|u|). Check that this is correct:

Now we can integrate both sides and simplify.

Here, we have written C₂ in place of e^C1, renaming the constant.

Again, we write C in place of (C₂)².

The solution is a parabola. The family of solution curves looks like this: