(Solution) - An alternative approach to determining stability is the direct methed -(2025 Original AI-Free Solution)

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An alternative approach to determining stability is the direct methed of Liapunov.5 Liapunov assumes the existence of a positive-definite energylike function L(x, y) with continuous first partial derivatives.6 His theorem states that if (0, 0) is an isolated equilibrium solution of x' = f(x, y) and y' = g(x, y), and if

(the derivative of L along the trajectory) is negative definite on a neighborhood of the origin, then the origin is asymptoticall y stable.
Use Liapunov's direct method to verify the asymptotic stability of the origin for each system in Problem, after checking that the given function L is a legitimate Liapunov function.
x' = y - 2x3
y' = - 2x - 3y5
L(x, y) = 2x2 + y2