Is $u \frac{dv}{du} + v (1+u) = u e^{u}$ a linear or nonlinear ODE? I am having a hard time deciding if the following differential equation is linear or not because of the last term on the left handside.

$$u \frac{dv}{du} + v (1+u) = u e^{u}$$

Can $v$ be multiplied by a linear function of $u$ like $(1+u)$ and still make the differential equation linear?

Thank you
Any equation first order of this form is linear:
$$y'+\alpha(x)y=\beta(x)$$
You DE is of that form
$$v'+\frac {u+1}{u}v=e^u
\implies v'+\alpha ({u})v=\beta(u)$$ 
So it's linear.
The solution is given by $$v(u)=\frac{c_1 e^{-u}}{u}+\frac{e^u \left(\frac{u}{2}-\frac{1}{4}\right)}{u}$$