Q10E Find all the critical points of ... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 5: Q10E (page 271)

Find all the critical points of the system
$\frac{d x}{d t} = x^{2} - 1 \frac{d y}{d t} = x y$
and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

Short Answer

Expert verified

The result is

$f o r 鈥� y > 0, |x| > 1, y = e^{c} \sqrt{x^{2} - 1} f o r 鈥� y > 0, |x| < 1, y = e^{c} \sqrt{1 - x^{2}} f o r 鈥� y < 0, |x| > 1, y = - e^{c} \sqrt{x^{2} - 1} f o r 鈥� y < 0, |x| < 1, y = - e^{c} \sqrt{1 - x^{2}}$

And the end points are (-1,0) (1,0).

Step by step solution

Find critical points

For a critical point put the system equal to 0.

$x^{2} - 1 = 0 x y = 0 x^{2} = 1 x = 卤 1 y = 0$

Find the value of y

Now,

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{x y}{x^{2} - 1} \\ 鈭� \frac{1}{y} d y & = & 鈭� \frac{x^{2}}{x^{2} - 1} \\ \ln |y| & = & \frac{1}{2} \ln |t| + c 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� (bysubtitution) \\ \ln |y| & = & \ln {|t|}^{\frac{1}{2}} + c \\ y & = & e^{c} {|t|}^{\frac{1}{2}} \\ y & = & e^{c} \sqrt{x^{2} - 1} \end{array}$

Prove results that endpoints are semicircles.

Now there are two cases when y > 0 and y < 0 for both cases $x^{2} - 1 < 0 鈥� 鈥� and 鈥� 鈥� x^{2} - 1 > 0$ then $|x| < 1 鈥� 鈥� a n d 鈥� 鈥� |x| > 1$

$f o r y > 0, |x| > 1, y = e^{c} \sqrt{x^{2} - 1} f o r y > 0, |x| < 1, y = e^{c} \sqrt{1 - x^{2}} f o r y < 0, |x| > 1, y = - e^{c} \sqrt{x^{2} - 1} f o r y < 0, |x| < 1, y = - e^{c} \sqrt{1 - x^{2}}$

The trajectories that possibly lie on semicircles. If we square the equation then

$y^{2} = e^{2 c} (1 - x^{2})$

This will be the equation of circle only if $e^{2 c} = 1$ . Therefore, only two solutions lie on the semicircle, $y = \sqrt{x^{2} - 1}, y = - \sqrt{x^{2} - 1}$ .

Therefore, it is a circle of radius 1 center at origin the endpoints are $(- 1, 0), 鈥� 鈥� (1, 0)$ .

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91影视

Find all the critical points of the system
$\frac{d x}{d t} = x^{2} - 1 \frac{d y}{d t} = x y$
and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

Short Answer

Step by step solution

Find critical points

Find the value of y

Prove results that endpoints are semicircles.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Find all the critical points of the systemdxdt=x2-1dydt=xyand the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

Short Answer

Step by step solution

Find critical points

Find the value of y

Prove results that endpoints are semicircles.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find all the critical points of the system
$\frac{d x}{d t} = x^{2} - 1 \frac{d y}{d t} = x y$
and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?