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Chapter 5: Q5.4-1E (page 271)

In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.

dxdt=3y3,dydt=y;x(t)=e3t,y(t)=et

Short Answer

Expert verified

By putting the values of xt,  yt, get the result.

Step by step solution

01

Get the result in form of x and y

Here the system is

dxdt=3y3dydt=y

And

xt=e3tyt=et

Then

role="math" localid="1663936766613" dxdt=3e3t=3yt3dydt=et=yt

Therefore, the pair is a solution to the system.

Also,yt3=xt=x13

02

Get the result

Sincedxdt=3y3x isincreasing when role="math" localid="1663936934373" y>0and x is decreasing for role="math" localid="1663936952067" y<0. This means the flow is from left to right along the part of the curve that lies above the x-axis, and the flow is from right to left along the part of the curve that lies below the x-axis.

03

Sketch the graph.

This is the required result.

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