## Homogeneous coefficients corner case

Before I started the program, I implemented Bernoulli equations. But the general solution to Bernoulli equations involves raising something to the power of $\frac{1}{1-n}$, where n is the power of the dependent term (see the Wikipedia page for more info). This works great, as I soon discovered, unless n == 1. Then you get something to the power of $\infty$. So I had to go in and remove the corner case.

So you think that after that I would have been more careful after that about checking that if general solution that divides by something I would test to see if that something is not zero before returning it as a solution.

Well, as I was just trying to implement some separable equation tests, I was going through the exercises of my ode text as I usually do for tests, and I came across $xy' - y = 0$. If you recall, this equation also has coefficients that homogeneous of the same order (1). From the general solution to homogeneous coefficients, you would plug it into $\int{\frac{dx}{x}}=\int{\frac{-Q(1,u)du}{P(1,u)+uQ(1,u)}}+C$ where $u = \frac{y}{x}$ or $\int{\frac{dy}{y}}=\int{\frac{-P(u,1)du}{uP(u,1)+Q(u,1)}}+C$ where $u = \frac{x}{y}$ (here, P and Q are from the general form $P(x,y)dx+Q(x,y)dy=0$). Well, it turns out that if you plug the coefficients from my example into those equations, the denominator will become 0 for each one. So I (obviously) need to check for that $P(1,u)+uQ(1,u)$ and $uP(u,1)+Q(u,1)$ are not 0 before running the homogeneous coefficients solver on a differential equation.