I have a very simple linear first order ODE.

$$v(x) = c x + A - B x(1-x) v'(x)$$

$c, A ,B \in(0,1)$. The domain is $(\underline{x}, 1)$. where $\underline x > 0$.

I am guessing that for any initial condition $v(\underline x)$ there will be a unique solution. In particular, suppose I give an initial condition $v(\underline x)$ such that $0 < v(\underline x) < c \underline x + A$. I would like to argue that the solution must be convex. What are the sort of methods that I can use when an explicit closed form solution isn't available?

Thanks