equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
An ordinary differential equation is a differential equation involving derivatives of a function with respect to one argument only, i.e. the function is on a manifold only of dimension $d = 1$. This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter.
A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let $X$ be a Banach space; denote the ball of radius $r$ about $x \in X$ by $B_r(x)$. Given $(t_0, x_0) \in \mathbb{R} \times X$ and $a, r \gt 0$, put $Q(a, r) \coloneqq [t_0 - a, t_0 + a] \times B_r(x_0)$.
(Picard-Lindelöf) Suppose $f: Q(a, b) \to X$ is a continuous function satisfying the following conditions:
(Lipschitz condition) There is a Lipschitz constant $L$ such that
for all $(t, x) \in Q(a, b)$;
(Boundedness) There is a constant $K$ such that $\sup_{(t, x) \in Q(a, b)} \|f(t, x)\| \leq K$.
Then for any $y_0 \in X$ and $c \coloneqq \min(a, b/K)$, there exists exactly one solution $y: [t_0 - c, t_0 + c] \to X$ to the initial value problem