Paper 2, Section II, A

(a) Find the general real solution of the system of first-order differential equations

$\begin{aligned} &\dot{x}=x+\mu y \\ &\dot{y}=-\mu x+y, \end{aligned}$

where $\mu$ is a real constant.

(b) Find the fixed points of the non-linear system of first-order differential equations

$\begin{aligned} &\dot{x}=x+y \\ &\dot{y}=-x+y-2 x^{2} y \end{aligned}$

and determine their nature. Sketch the phase portrait indicating the direction of motion along trajectories.

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