Homogeneous differential equations
The word “homogeneous” can mean different things depending on what kind of differential equation you’re working with. A homogeneous equation in this sense is defined as one where the following relationship is true:
The solution to a homogeneous equation is to:
- Use the substitution where is a substitution variable.
- Implicitly differentiate the above equation to get .
- Replace and with these expressions.
- Solve for .
- Substitute with the expression Then solve for .
The advantage of this method is that the function is in terms of 2 variables, but we simplify the equation by relating and to each other.