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Consider the nonlinear second-order differential equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant....

Consider the nonlinear second-order differential equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant. Answer to the following questions.
(a) Show that there is no periodic solution in a simply connected region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to Theorem 11.5.1>>
If symply connected region R either contains no critical points of plane autonomous system or contains a single saddle point, then there are no periodic solutions. )
(b) Derive a plane autonomous system from the given equation and find all the critical points.
(c) Classify(discriminate/categorize) all the critical points into one of the three categories: {stable / saddle / unstable(not saddle)}.

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