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Putnam 1973 A5

Source: Putnam 1973

May 25, 2022
Putnamtrigonometryphysicsdifferential equation3d

Problem Statement

A particle moves in 33-space according to the equations: dxdt=yz,  dydt=xz,  dzdt=xy. \frac{dx}{dt} =yz,\; \frac{dy}{dt} =xz,\; \frac{dz}{dt}= xy. Show that: (a) If two of x(0),y(0),z(0)x(0), y(0), z(0) equal 0,0, then the particle never moves. (b) If x(0)=y(0)=1,z(0)=0,x(0)=y(0)=1, z(0)=0, then the solution is x(t)=sec⁡t,  y(t)=sec⁡t,  z(t)=tan⁡t; x(t)= \sec t ,\; y(t) =\sec t ,\; z(t)= \tan t; whereas if x(0)=y(0)=1,z(0)=−1,x(0)=y(0)=1, z(0)=-1, then x(t)=1t+1,  y(t)=1t+1,z(t)=−1t+1. x(t) =\frac{1}{t+1} ,\; y(t)=\frac{1}{t+1}, z(t)=- \frac{1}{t+1}. (c) If at least two of the values x(0),y(0),z(0)x(0), y(0), z(0) are different from zero, then either the particle moves to infinity at some finite time in the future, or it came from infinity at some finite time in the past (a point (x,y,z)(x, y, z) in 33-space "moves to infinity" if its distance from the origin approaches infinity).
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