y' + y/x = q(x)
It is first order, linear, non-constant coefficients, inhomogeneous.
I suspect an integrating factor is required, but I am not quite sure how the process works.
$\displaystyle y'+y*p(x)=q(x)$
The integrating factor will be $\displaystyle e^{\int p(x)dx}$ as is always the case with equations of this form
$\displaystyle \int\frac{1}{x}dx=ln|x|$ and $\displaystyle e^{ln|x|}=x$
Yes we can ignore the absolute value
Now we multiply both sides by the integrating factor (x in this case)
$\displaystyle xy'+y=x*q(x)\rightarrow (xy)'=x*q(x)$ and we integrate both sides with respect to x to get
$\displaystyle \int (xy)'dx=\int x*q(x)dx\rightarrow xy=\int x*q(x)dx$
So $\displaystyle y=\frac{1}{x}\int x*q(x)dx$
You do not need a constant of integration for the integrating factor
You do need one for the final step if you knew what q(x) was because that is how you get the family of functions that satisfy the differential equation
If you had an initial value, then you could solve for the constant explicity after you've integrated, but yes, you need a +C (or some constant) on the final integration to get the family of answers and not just one specific function