Integral confusion for a simple Differential Equation

[anthraxiom]

TL;DR

Say there is a function y where dy/dx=y/x. Now if we rearrange we get dy/y= dx/x. integrating both sides, we get lny=lnx. , x=y

I simply don't know where I'm going wrong in this. lets for example say y=2x. dy/dx=y/x=2
now if we look at only the differential equation we see that dy/y=dx/x, solving we get x=y
I have no idea how this is happening, please , if possible guide my foolish thoughts to where I have gone wrong.

 

[pasmith]

You are forgetting the constant of integration. $$\int \frac{dy}{y} = \int \frac{dx}{x} \Rightarrow \ln |y| = \ln |x| + \ln |A| \Rightarrow y = Ax.$$

 

[anthraxiom]

You are forgetting the constant of integration. $$\int \frac{dy}{y} = \int \frac{dx}{x} \Rightarrow \ln |y| = \ln |x| + \ln |A| \Rightarrow y = Ax.$$

thanks :D