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Chapter 5: Q13E (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=(y-x)(y-1)dydt=(x-y)(x-1)

Short Answer

Expert verified

The solution is(y-1)2+(x-1)2=c.

Step by step solution

01

Find phase plane equation

Here the system is:

dxdt=(y-x)(y-1)dydt=(x-y)(x-1)

And the phase plane equation is:

dydx=(x-y)(x-1)(y-x)(y-1)dydx=1-xy-1

02

Solve the equation

Here the equation is dydx=1-xy-1.

Solving by variable separating. Then,

鈭�y-1dy=鈭�-x-1dxy-12+x-12=c

Since the solutions are centered at (1,1). And line y = x.

03

Sketch some trajectories.

Therefore, the solution is (y-1)2+(x-1)2=c.

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Most popular questions from this chapter

Figure 5.16 displays some trajectories for the system dxdt=y,dydt=-x+x2What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?

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.

dxdt+x+dydt=e4t,2x+d2ydt2=0

Solve the given initial value problem.

x'=z-y;x(0)=0y'=z;鈥�y(0)=0z'=z-x;z(0)=2

In Problems 29 and 30, determine the range of values (if any) of the parameter that will ensure all solutions x(t), and y(t) of the given system remain bounded as t鈫�+鈭�.

dxdt=-x+位测,dydt=x-y

In Problems 10鈥�13, use the vectorized Euler method with = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

y''+ty'+y=0;y(0)=1,y'(0)=0on[0,1]
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