State the Lyapunov instability theorem and write its proof. Then using it, show the instability of the equilibrium point (0, 0) of the following differential equation.
x^(?)=x+y+f(x,y),y^(?)=-x+y+g(x,y)
where, in a small neighborhood of the origin, the functions f and g have the following properties.
(|f(x,y)|)/(||x,y||)<=\alpha ||x,y||,(|g(x,y)|)/(||x,y||)<=\alpha ||x,y||,\alpha >0
Hint: Don't forget to use the Lyapunov equation to find the Lyapunov function.

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State the Lyapunov instability theorem and write its proof. Then using it, show the instability of the equilibrium point (0, 0) of the following differential equation.

x^(?)=x+y+f(x,y),y^(?)=-x+y+g(x,y)

where, in a small neighborhood of the origin, the functions f and g have the following properties.


(|f(x,y)|)/(||x,y||)<=\alpha ||x,y||,(|g(x,y)|)/(||x,y||)<=\alpha ||x,y||,\alpha >0

Hint: Don't forget to use the Lyapunov equation to find the Lyapunov function.

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