Linearization of nonlinear differential equations near an equilibrium position

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. 


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Erdos Erdos
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