How do you solve dy/dx=y/x using homogeneous substitution?

Question

I am aware the answer is y=Ax where A is a constant, and I know how to solve it using separable variables and integrating factor. I would like to know how to solve it specifically through the method homogeneous substitution and get the same answer.

Answer 1

Hope, the following is what you need.

Substitute: y = xu

Then dy = x du + u dx

dy/dx = x du/dx + u, y/x = u

The new equation: x du/dx + u = u

x du/dx = 0

x can be equal to any value, so du/dx = 0

u = A (a constant)

So, the solution y = Ax