How to solve dy/dx=y/x using homogeneous equation if the answer is y=Ax where A is a constant?

Answer 1

Given the information in the problem, let's try to "separate" variables so that only one side has the y variable and the other side has the x variable in this equality.

dy/dx = y/x

1/y * dy/dx = 1/y * y/x [ y cannot be zero ]

1/y * dy/dx = 1/x

dx * 1/y * dy/dx = dx * 1/x

Simplifying...

dy/y = dx/x

Now that separation is completed, we can integrate both sides of the quality to remove the differentials...

We have an indefinite integral due to no boundary information for the integration...

ln(y) = ln(x) + C

e^ln(y) = e^{(ln(x)+ C)}

y = e^C * x

Let e^C = A

y = Ax

We can now relax the restriction of y not equal to zero due to the 0/0 issue we would have encountered from the original equation.