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Chapter 5: Q18E (page 271)

In Problems 15–18, find all critical points for the given system. Then use a software package to sketch the direction field in the phase plane and from this description the stability of the critical points (i.e., compare with Figure 5.12).

dxdt=x(7-x-2y),dydt=y(5-x-y)

Short Answer

Expert verified

This result is the unstable node point (0,0),stable node (7,0),stable node (0,5) and saddle point (3,2).

Step by step solution

01

Find critical points

Here the system is;

dxdt=x(7-x-2y)dydt=y(5-x-y)

For critical points equate the system equal to zero.

x(7-x-2y)=0y(5-x-y)=0

Solve for x and y get the four points (0,0),(7,0),(0,5),(3,2).

So, this is the unstable node point (0, 0), stable node (7, 0), stable node (0, 5) and saddle point (3, 2).

02

Sketch

This is the required result.

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Most popular questions from this chapter

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.

y''(t)=cos(t-y)+y2(t);y(0)=1,y'(0)=0

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.

y6(t)=y'(t)3-sin(y(t))+e2t;y(0)=y'(0)=......=y(5)(0)=0

In Problems 25 – 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), and z(t).

x'=x+2y+z,y'=6x-y,z'=-x-2y-z

In Problems 25 – 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), z(t).

x'=3x+y-z,y'=x+2y-z,z'=3x+3y-z

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
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