Topological argument of differential equation and solution space. In physics textbooks, when $\sin{\theta}$ or $\cos{\theta}$ appears in a differential equation, it is often approximated using Maclaurin series.
It's not trivial for me that when we replace a function, for example, $\sin{\theta}$, with a function which is "close" to that, for example, $\theta$, the solution of original differential equation and new differential equation are also "close" (and how large the error is).
I would like to know general argument of this kind.
 A: I know the following general theorem [P, Th. 13], which can be helpful.
Consider a system 
$$\frac {dx^i}{dt}=f^i(t,x^1,\dots, x^n,\mu^1,\dots,\mu^l), \,\, i=1,\dots, n.$$
In this system $t$ is the independent variable, $x^1,\dots, x^n$ are unknown functions of this variable, the right hand sides $f^i$ of the system depends on parameters $\mu^1,\dots,\mu^l$ and are defined on an open set  $\tilde\Gamma$ of the space $\tilde R$ of variables $t,x^1,\dots, x^n,\mu^1,\dots, \mu^l$. We shall assume that both the functions $f^i$ and their partial derivatives $\frac{\partial f^i}{\partial x^j}$ for all $i, j=1,\dots, n$ are (jointly) continuous on $\tilde\Gamma$.
A point of the space $\tilde R$ we shall denote by $(t,x,\mu)$. Fix initial values $t_0,x_0$ and denote by $M$ the set of all $\mu$ such that a point $(t_0,x_0,\mu)$ belongs to $\tilde\Gamma$. Clearly, $M$ is an open set in the space of variables $\mu^1,\dots,\mu^l$. To each point $\mu$ of the set $M$ corresponds a non-extendable solution $\varphi(t,\mu)$ with the initial  conditions $t_0,x_0$ of the system, defined on an interval $m_1(\mu)<t<m_2(\mu)$. A set $T$ of all pairs $(x,\mu)$ for which the function $\varphi(t,\mu)$ is determined is described by the following conditions: a point $\mu$ belongs to the set $M$ and $m_1(\mu)<t<m_2(\mu)$. Theorem 13 claims that the set $T$ is open in the space of variables $t, \mu^1,\dots,\mu^l$ and the function $\varphi(t,\mu)$ is (jointly) continuous on $T$.
References
[P] L. S. Pontrjagin, Usual differential equations, 4th edition, Moskow: Nauka, 1974 (in Russian) 
