Paper 2, Section II, A

Differential Equations | Part IA, 2012

Consider the function

V(x,y)=x4x2+2xy+y2V(x, y)=x^{4}-x^{2}+2 x y+y^{2}

Find the critical (stationary) points of V(x,y)V(x, y). Determine the type of each critical point. Sketch the contours of V(x,y)=V(x, y)= constant.

Now consider the coupled differential equations

dxdt=Vx,dydt=Vy\frac{d x}{d t}=-\frac{\partial V}{\partial x}, \quad \frac{d y}{d t}=-\frac{\partial V}{\partial y}

Show that V(x(t),y(t))V(x(t), y(t)) is a non-increasing function of tt. If x=1x=1 and y=12y=-\frac{1}{2} at t=0t=0, where does the solution tend to as tt \rightarrow \infty ?

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