Consider the differential equation $x ^ { \prime } = \sin x + \cos x$ . The point $x = 3 \pi / 4$ is
A) a stable critical point
B) an unstable critical point
C) not a critical point

Question

Quizplus · Accepted Answer

The point $x = 3 \pi / 4$ is a critical point because $x ^ { \prime } ( 3 \pi / 4 ) = 0$. To determine its stability, we linearize the equation around this point:$$x ^ { \prime } = \sin x + \cos x \approx \sin ( 3 \pi / 4 ) + \cos ( 3 \pi / 4 ) ( x - 3 \pi / 4 ) = -1/ \sqrt { 2 } - ( x - 3 \pi / 4 ) / \sqrt { 2 }$$The linearized equation is $x ^ { \prime } = - ( x - 3 \pi / 4 ) / \sqrt { 2 }$, which has a negative slope. Therefore, the critical point $x = 3 \pi / 4$ is stable.

Consider the Differential Equation x′=sin⁡x+cos⁡xx ^ { \prime } = \sin x + \cos xx′=sinx+cosx

Consider the Differential Equation $x ^ { \prime } = \sin x + \cos x$