Chapter 2.2, Problem 12P

Program Plan Intro

Program Description: Purpose ofproblem is to findthe critical point of the differential equation $\frac{dx}{dt}={\left(2-x\right)}^3$ and then check whether the critical point is stable or unstable and further solve the differential equation in explicit way for $x\left(t\right)$ in terms of $t$ and then sketch the typical solution curves for the differential equation to find exact solution.

Summary introduction:

Problem will use the separation of the variable for differential equation $\frac{dy}{dx}=f\left(x\right)g\left(y\right)$ and the integral of $x^{-1}$ can be calculated as ${\displaystyle \int \frac{1}{x}dx=\ln \left|x\right|+c}$ where $c$ is called the constant of integration.