Solving the Differential Equation
The given differential equation is:
xy² + x²y = 0
To solve this equation, we can start by factoring it:
y(xy + x²) = 0
This equation gives us two factors that can be set to zero:
- Factor 1: y = 0
- Factor 2: xy + x² = 0
Case 1: y = 0
This case clearly shows that y is equal to zero for all values of x, which is one solution of the differential equation.
Case 2: xy + x² = 0
We can rewrite this factor:
xy = -x²
Now, divide both sides by x (assuming x ≠ 0):
y = -x
This gives us another solution where y is a linear function of x, specifically, y equals negative x.
General Solution
Therefore, the general solution to the differential equation xy² + x²y = 0 includes both cases:
- y = 0
- y = -x
In conclusion, the solutions are:
y = 0
or
y = -x