Paper 2, Section II, C

Differential Equations | Part IA, 2019

Consider the nonlinear system

x˙=y2y3y˙=x\begin{aligned} &\dot{x}=y-2 y^{3} \\ &\dot{y}=-x \end{aligned}

(a) Show that H=H(x,y)=x2+y2y4H=H(x, y)=x^{2}+y^{2}-y^{4} is a constant of the motion.

(b) Find all the critical points of the system and analyse their stability. Sketch the phase portrait including the special contours with value H(x,y)=14H(x, y)=\frac{1}{4}.

(c) Find an explicit expression for y=y(t)y=y(t) in the solution which satisfies (x,y)=(12,0)(x, y)=\left(\frac{1}{2}, 0\right) at t=0t=0. At what time does it reach the point (x,y)=(14,12)?(x, y)=\left(\frac{1}{4},-\frac{1}{2}\right) ?

Typos? Please submit corrections to this page on GitHub.