2.II.7B

Show how a second-order differential equation $\ddot{x}=f(x, \dot{x})$ may be transformed into a pair of coupled first-order equations. Explain what is meant by a critical point on the phase diagram for a pair of first-order equations. Hence find the critical points of the following equations. Describe their stability type, sketching their behaviour near the critical points on a phase diagram.

$\begin{aligned} &\text { (i) } \ddot{x}+\cos x=0 \\ &\text { (ii) } \ddot{x}+x\left(x^{2}+\lambda x+1\right)=0, \quad \text { for } \lambda=1,5 / 2 . \end{aligned}$

Sketch the phase portraits of these equations marking clearly the direction of flow.

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