I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example:
$ y^{2}(t) + y(t) = \sin(t)$
I am coming from a signal processing background, so I am trying to view this in terms of an input-output relationship; so $y(t)$ is the unknown output, and $\sin(t)$ is the known input or forcing function. It is possible to represent this as a signal flow diagram that represents the equation $y(t) = x(t) - f(y)$.
I realize that this looks like a simple feedback system, but I am specifically interested in examining the situation where the system is purely algebraic, not the standard systems described by differential equations.
I am not aware of a way to solve such equations, and also do not know what field of math this falls under (Algebraic Geometry?). Any guidance or recommendations would be greatly appreciated.