# Solving a non-linear parametric equation

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.

• This may seem naive, but have you considered using the quadratic formula to obtain a closed form expression for $y(t)$? Jan 7, 2016 at 23:33
• Could you elaborate a bit? It wouldn't work for the general case, but I'm curious what you mean. Jan 7, 2016 at 23:36
• If one considers $y(t)$ as an "unknown" in the first equation you wrote, this is a quadratic equation and we can write: $y(t)= \frac{1}{2}(-1 \pm \sqrt{1+4\sin (t)})$. This clearly won't be real everywhere, but I don't know if you need an everywhere-real solution. Jan 7, 2016 at 23:39
• Ah, I hadn't considered that, but it makes sense. Any idea if approaching these problems in terms of signals has been done? Jan 7, 2016 at 23:42
• I don't know, that's why I didn't put this as an answer. I know very little about signal processing. Jan 8, 2016 at 1:09