How do you solve the differential equation xy² + x²y = 0?

To solve the differential equation xy² + x²y = 0, we start by factoring the left-hand side. This gives us:

y(xy + x²) = 0

This equation consists of two factors that can each equal zero. Thus, we can break it down into two separate cases:

1. Case 1: y = 0
   In this case, the solution is simply y = 0, which is a valid solution for all x.
2. Case 2: xy + x² = 0
   Simplifying this gives us y = -x. This is the second solution to the equation.

Combining both solutions, the general solution of the differential equation is:

y = 0 or y = -x

In conclusion, we have two solutions derived from our analysis of the given differential equation.