Consider the differential equation $x ^ { \prime } = \tan x$ . The point $x = 0$ is
A) a stable critical point
B) an unstable critical point
C) not a critical point

Question

Quizplus · Accepted Answer

A critical point, or equilibrium point, of a differential equation $x' = f(x)$ is a point where $f(x) = 0$. For the given equation $x' = \tan(x)$, $x = 0$ is a critical point because $\tan(0) = 0$. To determine stability, we look at the derivative of $f(x)$ with respect to $x$, which is $f'(x) = \sec^2(x)$. At $x = 0$, $f'(0) = \sec^2(0) = 1$, which is positive, indicating that the critical point at $x = 0$ is unstable.

Consider the Differential Equation x′=tan⁡xx ^ { \prime } = \tan xx′=tanx

Consider the Differential Equation $x ^ { \prime } = \tan x$