Differential equations/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:

${\displaystyle \textstyle f(tx,ty)=t\cdot f(x,y)}$

The solution to a homogeneous equation is to:

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

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