Oh, perhaps it's separable?
So then I get
But how do I integrate the LHS? Partial fraction expansion? Anything easier?
The Differential equation is a special equation called the Riccati equation. It reduces to the Bernoulli's equation for
The solution of this equation is not very simple. One method is to make the substitution
where u(t) is a particular solution that is already known,
y is a new dependent variable.
This substitution reduces the equation to a Bernoulli's equation in y and t, and you can obtain y as a function of t. Returning to the original dependent variable gives you the actual solution as
In this case, you can take the particular solution to be one of the roots of the equation wherever is real.
The solution will depend on since might have:
Case 1: two distinct linear factors,
Case 2: one repeated linear factor, or
Case 3. no real linear factor
depending on the value of . For case 1, use partial fractions. For case 2, it's a standard form. For case 3 you need to complete the square and then recognise a standard form that will give you arctan.
The work hack work is left for you. (You might find it easier to start with concrete values for for each case. Note that there is only one value for case 2).