How to determine the effect of a singularity of a function entering the ODE on the solution?

Question

Let $R:\mathbb{R}\to\mathbb{R}$ a given function and let us consider the following Riccati differential equation: $$ f''(t)+f'(t)^2-R(t)=0. $$ My question is: Is there any theory (please, provide a book, paper whatever) which allows to determine what is the effect of a discontinuity of the function $R(t)$ on the solution of the ODE? I am also interested in understanding whether a change of concavity of the function produces a well-determined effect on the solution of the ODE.

Answer 1

Without any further properties on that $R(t)$ I think that there's not too much to say. I would say that, in general, questions specifics like this are solved with a combination of well know inequalities, bounds, etc and some "exploration" over the, a priori selected, particular properties of the coefficients. Take a look on this paper for example of what I'm saying.

Suppose that the kind of function $R(t)$ has a finite discontinuity like the sign function. So your equation becomes: $$y'=-y^2+sgn(x)$$ So what you really have is a vector field in $R^2$ that is obtained by gluing together two smooth vector fields: $$x'=1$$ $$y'=-y^2-1$$ for $(x,y)$ with $x\le0$, and $$x'=1$$ $$y'=-y^2+1$$ for $(x,y)$ with $x>0$. And, in this case, the effect of the discontinuity will be the appearance of a sharp corner in the solutions.