I have a the following Riccatti type ODE defined in $t\in [0,T]$ . I am quite sure this does not allow a solution in terms of elementary functions.
$$\frac{dy}{dt}=y(2+\hat{v}-0.15t)-\frac{1}{2}y^2-1$$ with the following boundary conditions $y(T)=0$ and $y(0)=\hat{v}$
I am very new to Boundary value problems. I was able to solve this numerically using a shooting method by guessing a $\hat{v}$ and solving the corresponding IVP till the the error is small enough. I was wondering how I can say that such a procedure will always lead to a solution.
Is there any way that I argue that that a solution to this BVP will always exist?