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Chapter 5: Q14E (page 271)

In Problems 11鈥�14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

dxdt=3ydydt=2x

Short Answer

Expert verified

The solution is y=cx23.

Step by step solution

01

Find phase plane equation

Here the system is;

dxdt=3ydydt=2x

And the phase plane equation is;

dydx=2y3x

02

Solve the equation

Here the equation is dydx=2y3x.

Solving by variable separating. Then

鈭�1ydy=鈭�23xdxlny=lnx23+cy=cx23

03

Sketch some trajectories

This is the required result.

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Most popular questions from this chapter

In Problems 7鈥�9, solve the related phase plane differential equation (2), page 263, for the given system.

dxdt=y-1,dydt=ex+y

Arms Race. A simplified mathematical model for an arms race between two countries whose expenditures for defense are expressed by the variables x(t) and y(t) is given by the linear system

dxdt=2y-x+a;鈥�鈥�鈥�鈥�x0=1,dydt=4x-3y+b;鈥�鈥�鈥�y0=4,

Where a and b are constants that measure the trust (or distrust) each country has for the other. Determine whether there is going to be disarmament (x and y approach 0 as t increases), a stabilized arms race (x and y approach a constant ast鈫�+鈭� ), or a runaway arms race (x and y approach+鈭� as t鈫�+鈭�).

In Problems 3鈥�6, find the critical point set for the given system.

dxdt=x2-2xy,dydt=3xy-y2

In Problems 3 鈥� 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.
x'=x-y,y'=y-4x

Solve the given initial value problem.

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