Chapter 5: Q13E (page 271)

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).
$\frac{dx}{dt} = (y - x) (y - 1) \frac{dy}{dt} = (x - y) (x - 1)$

Short Answer

Expert verified

The solution is $(y - 1)^{2} + (x - 1)^{2} = c$ .

Step by step solution

Find phase plane equation

Here the system is:

$\frac{d x}{d t} = (y - x) (y - 1) \frac{d y}{d t} = (x - y) (x - 1)$

And the phase plane equation is:

$\frac{d y}{d x} = \frac{(x - y) (x - 1)}{(y - x) (y - 1)} \frac{d y}{d x} = \frac{1 - x}{y - 1}$

Solve the equation

Here the equation is $\frac{d y}{d x} = \frac{1 - x}{y - 1}$ .

Solving by variable separating. Then,

$\int (y - 1) d y = \int - (x - 1) d x {(y - 1)}^{2} + {(x - 1)}^{2} = c$

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

Sketch some trajectories.

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

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In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).
$\frac{dx}{dt} = (y - x) (y - 1) \frac{dy}{dt} = (x - y) (x - 1)$

Short Answer

Step by step solution

Find phase plane equation

Solve the equation

Sketch some trajectories.

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In Problems 11–14, 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

Step by step solution

Find phase plane equation

Solve the equation

Sketch some trajectories.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Calculus

Statistics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).
$\frac{dx}{dt} = (y - x) (y - 1) \frac{dy}{dt} = (x - y) (x - 1)$