Answer
This differential equation is linear.
Work Step by Step
A first-order linear differential equation is one that can be put into the form:
$\frac{dy}{dx} + P(x)~y = Q(x)$
We can consider the given equation:
$ue^t = t+ \sqrt{t}~\frac{du}{dt}$
$\frac{ue^t}{\sqrt{t}} = \frac{t}{\sqrt{t}}+ \frac{du}{dt}$
$\frac{du}{dt}+ \sqrt{t}= \frac{ue^t}{\sqrt{t}}$
$\frac{du}{dt} + (-\frac{\sqrt{t}~e^t}{t})~u = -\sqrt{t}$
Let $P(t) = -\frac{\sqrt{t}~e^t}{t}$
Let $Q(t) = -\sqrt{t}$
This differential equation is linear.