The differential equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x e ^ { 0.01 x } = 0$ is a model for an undamped spring-mass system with a nonlinear restoring force. The initial conditions are $x ( 0 ) = 0$ , $x ^ { \prime } ( 0 ) = 1$ . The solution of the linearized system is
A) $x = \left( e ^ { t } + e ^ { - t } \right) / 2$
B) $x = \left( e ^ { t } - e ^ { - t } \right) / 2$
C) $x = \cos t$
D) $x = \sin t$
E) $x = \cos t - \sin t$

Question

Quizplus · Accepted Answer

To linearize the given differential equation, we ignore the nonlinear term $e^{0.01x}$ and approximate it as 1 when $x$ is near 0. This simplifies the equation to $\frac{d^2x}{dt^2} + x = 0$, which is the equation for a simple harmonic oscillator. The solution that satisfies the initial conditions $x(0) = 0$ and $x'(0) = 1$ is $x = \sin t$.

The Differential Equation d2xdt2+xe0.01x=0\frac { d ^ { 2 } x } { d t ^ { 2 } } + x e ^ { 0.01 x } = 0dt2d2x​+xe0.01x=0

The Differential Equation $\frac { d ^ { 2 } x } { d t ^ { 2 } } + x e ^ { 0.01 x } = 0$