# Homogeneous Differential Equation

A first order equation $\frac{dy}{dx} = f(x,y)$ is said to be a homogenous equation if $f(x,y) = f(tx,ty)$.

Steps to solve:

1. Substitute $y = vx$ into f(x,y) to get $\frac{dy}{dx} = f(x,vx)$
2. Differentiate $y = vx$ to get $\frac{dy}{dx} = x\frac{dv}{dx} + v$
3. Substitute the equation you got in (1) into (2).
4. Re-classify the differential equation from (3) and solve.
5. Substitute $v = \frac{y}{x}$ into the implicit solution from (4).
6. You’re done!

Back To First Order Differential Equations

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