The differential equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x \cos x = 0$ is a model for an undamped spring-mass system with a nonlinear forcing function. The initial conditions are $x ( 0 ) = 0.1$ , $x ^ { \prime } ( 0 ) = - 0.1$ . The solution of the linearized system is
A) $x = 0.1 \cos t$
B) $x = 0.1 \sin t$
C) $x = 0.1 \cos t - 0.1 \sin t$
D) $x = 0.1 e ^ { - t }$
E) $x = 0.1 e ^ { t }$

Question

Quizplus · Accepted Answer

To linearize the given differential equation, we approximate $\cos x \approx 1$ for small $x$, simplifying the equation to $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x = 0$. The solution to this linearized equation, given the initial conditions $x(0) = 0.1$ and $x'(0) = -0.1$, is $x = 0.1 \cos t - 0.1 \sin t$.

The Differential Equation d2xdt2+xcos⁡x=0\frac { d ^ { 2 } x } { d t ^ { 2 } } + x \cos x = 0dt2d2x​+xcosx=0

The Differential Equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x \cos x = 0$