We have , which is a first-order homogeneous differential equation.
It can be solved by rearranging to y dy=x dx and then integrating both parts which yields that .
Now if we use the substitution , and rewrite the differential equation as
and then rearrange to
by integrating both parts we get that
For (a special solution for c=0) , and by plugging into (1) we get that
What does equation (2) mean? is undefined. Is this of any significance?
As pointed out when rearranging from to , we implicitly assumed that . Equation (1) does not hold for
Solving equation (1) for u with , we arrive at the same family of equations but with . The fact that c can be zero comes from setting