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To solve the following d.e.:

xdy-ydx = 0 1

you get

ln(y)-ln(x)=C 2

no clearly you can simplify to:

ln(y/x)=c 3

which after taking exponentials gives:

y=Ax where A=e^c 4

however what interests me is if you do not simplify to

ln(y/c) in line 3 but simply take the exponential of line 2

then you get

y-x=A

obviously this is incorrect but I would like to know where I have gone wrong in the step above and also if it possible to get to the correct answer without using the simplification in step 3 above