How do you solve the following differential equation (Proof of Lemma 3 of Hermalin, 1998)?

Question

$(e(\theta) - s \theta)e'(\theta) = s(1-s)\theta$

The solution to the differential equation is given by: $e(\theta)=\frac{1}{2}(s+\sqrt{4s-3s^2})\theta$

Here, the dependent variable is $e(\theta)$, while $s$ can be treated like a constant.

Answer 1

The equation is homogeneous in the equal degrees sense, thus set $e(θ)=θv(θ)$ to get $$ θ(v-s)(θv'+v)=s(1−s)θ\implies (v-s)θv'=s(1−s)-v(v-s)\\~\\ \frac{(v-s)dv}{(v-\frac s2)^2+\frac34s^2-s}=-\frac{dθ}{θ}. $$ Now apply a separation in logarithmic and arcus tangent derivative on the left side and integrate, then try to solve for $v$.