Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

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.
$\frac{dx}{dt} = 3 y^{3}, \frac{dy}{dt} = y; x (t) = e^{3 t}, y (t) = e^{t}$

By putting the values of $x (t), y (t)$ , get the result.

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Step by Step Solution

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TABLE OF CONTENTS

Get the result in form of x and y

Here the system is

$\frac{d x}{d t} = 3 y^{3} \frac{d y}{d t} = y$

And

$x (t) = e^{3 t} y (t) = e^{t}$

Then

role="math" localid="1663936766613" $\frac{d x}{d t} = 3 e^{3 t} = 3 {(y (t))}^{3} \frac{d y}{d t} = e^{t} = y (t)$

Therefore, the pair is a solution to the system.

Also, ${(y (t))}^{3} = x (t) = x^{\frac{1}{3}}$

Get the result

Since $\frac{d x}{d t} = 3 y^{3}$ x is increasing when role="math" localid="1663936934373" $y > 0$ and x is decreasing for role="math" localid="1663936952067" $y < 0$ . This means the flow is from left to right along the part of the curve that lies above the x-axis, and the flow is from right to left along the part of the curve that lies below the x-axis.

Sketch the graph.

This is the required result.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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.
$\frac{dx}{dt} = 3 y^{3}, \frac{dy}{dt} = y; x (t) = e^{3 t}, y (t) = e^{t}$

Step by Step Solution

Get the result in form of x and y

Get the result

Sketch the graph.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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

Step by Step Solution

Get the result in form of x and y

Get the result

Sketch the graph.

Most popular questions for Math Textbooks

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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.
$\frac{dx}{dt} = 3 y^{3}, \frac{dy}{dt} = y; x (t) = e^{3 t}, y (t) = e^{t}$