# Differential equations/Homogeneous differential equations

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## 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:

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

### Solution

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.