Solving a complex ODE I'd like to solve the following ODE:
$$u'(z)+\frac{z-i}{z(z+i)}u(z)=g(z),$$
where $u$ is the unknown defined on complex domain and $g$ is a known smooth equation. I've tried to solve the homogenous part, but I don't know what contour should I integrate on. Furthermore, I can't find a specific solution to the inhomogenous part. Can anyone give a detailed solving process on this equation?

Multiplying each side of the equation by the integrating factor $\frac{(z+i)^2}{z}$ results in
$$\frac{(z+i)^2}{z}u'+\frac{(z+i)(z-i)}{z^2}u=\frac{(z+i)^2}{z} g(z), $$
and thus
$$\frac{d}{dz}\left(\frac{(z+i)^2}{z}u\right) = \frac{(z+i)^2}{z} g(z). $$
Therefore
$$u = \frac{z}{(z+i)^2} \int_a^z \frac{(s+i)^2}{s} g(s)\, \mathrm{d}s $$
for some constant $a\in \mathbb{C}$.
