Let $ t $ to be the independent variable and $ x(t) $ to be the unknown I want to solve the ODE for. Consider the following nonlinear nonhomogeneous ODE governing the dynamics of a real physical system:
$ f_1(k\dot{x}(t))\sin(x(t)) + f_2(k\dot{x}(t))\cos(x(t)) + f_3(k\dot{x}(t)) = g(t) $
where k is some scalar.
So it's a first order equation in $x(t)$, but as you see the functions $f_i$ are dependent on the first derivative.
(Or it could be a functional differential equation?)
Now my main questions are:
1- What are the conditions for existence and uniqueness?
2- How can I go about numerically solving this and what are the initial values I need? (is it just $x(0)$ or, $x(0)$ and $\dot{x}(0)$?)
I presume $f_1$, $f_2$, $f_3$ and $g$ are given smooth functions and $k \ne 0$. We want to consider the equation $$ F(p,x,t) = f_1(kp) \sin(x) + f_2(kp) \cos(x) + f_3(kp) - g(t) = 0$$ as implicitly defining $p$ as a differentiable function of $x$ and $t$. The Implicit Function Theorem says this is true at least locally if $\partial F/\partial p = k (f_1'(kp) \sin(x) + f_2'(kp) \cos(x) + f_3'(kp) \ne 0$. So as long as that condition holds we should have the standard local existence and uniqueness theorem.
In Maple you could solve this numerically as a DAE (differential-algebraic initial value problem) for $x(t)$ and $p(t)$.
Since it's a first-order equation you should normally require the initial condition on $x(0)$, but if there is more than one solution of $F(p,x(0),0)=0$ you'll also need to specify which of those is $\dot{x}(0)$.