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:

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

Solution

The solution to a homogeneous equation is to:

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

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