Question

7. Find a linearization of the differential equation in Problem 4.

Short answer

The linearization for the given differential equation can be reached using the Macluarin series of $e^{0.01x}$, under the condition that $x$is small.

Step-by-step solution

To Find the differential equation

We have the nonlinear second order differential equation

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

Our aim to put the given differential equation in the linear form. The Maclaurin series of the exponential function is defined as follows

$$\begin{gathered} 1+x+\frac{x^2}{2}+L \\ {e}^{0.01x}==1+(0.01x)+\frac{{(0.01x)}^2}{2}+L \\ {e}^{0.01x}\simeq 1\dots \left(1\right)\left[Forsmallvaluesofx\right] \end{gathered}$$

Final Answer

We substitute (l) into the given differential equation, yields

$\frac{d^{2}x}{d{t}^2}+x=0$

Which is a linear second order differential equation.

The linearization for the given differential equation can be reached using the Macluarin series of $e^{0.01x}$, under the condition that x is small.