$Linearization of nonlinear differential equations near an equilibrium position$

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Consider the nonlinear differential equation
\[\frac{d^2 x}{dt^2}=x-x^2.\]
(a) Show that $x=1$ is an equilibrium position for this equation.
(b) Find the linearized equation near $x=1$, and determine the stability of this equilibrium position.

Differential Equations Ordinary Differential Equations

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(a) To find the critical points,
\[x-x^2=0 \Rightarrow x(1-x)=0 \Rightarrow x=0,1.\]
So $x=1$ is a critical point.

(b) f(x)=x-x^2. So $f'(x)=1-2x$, and hence $f'(1)=-1$. Hence
\[f(x)=f(1)+f'(1)(x-1)+ \dots=0-1(x-1)+\dots =-(x-1)+\dots,\]
where $\dots$ represents higher order nonlinear terms. So the linearized equation is
\[\frac{d^2x}{dt^2}=-(x-1),\]
which represent a simple harmonic motion, and therefore $x=1$ is a stable equilibrium position.

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$Linearization of nonlinear differential equations near an equilibrium position$

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