Differential equations/Homogeneous differential equations

Homogeneous

Definition

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:

$f(tx,ty)=t\cdot f(x,y)$

Solution

The solution to a homogeneous equation is to:

1. Use the substitution $y=ux$ where $u$ is a substitution variable.
2. Implicitly differentiate the above equation to get $\frac{dy}{dx}=x\frac{du}{dx}+u$.
3. Replace $\frac{dy}{dx}$ and $y$ with these expressions.
4. Solve for $u$.
5. Substitute with the expression $u=\frac{y}{x}$ Then solve for $y$.

The advantage of this method is that the function is in terms of 2 variables, but we simplify the equation by relating $y$ and $x$ to each other.