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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