Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q24E

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 250

Book edition 9th

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

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.
$D [x] + (D + 1) [y] = e^{t}, D^{2} [x] + (D^{2} + D) [y] = 0$

The given system of equations is degenerate.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: General form

Elimination Procedure for 2 × 2 Systems:

To find a general solution for the system;

$L_{1} [x] + L_{2} [y] = f_{1}, L_{3} [x] + L_{4} [y] = f_{2}$

Where $L_{1}, L_{2}, L_{3},$ and $L_{4}$ are polynomials in $D = \frac{d}{dt}$ :

Make sure that the system is written in operator form.

Eliminate one of the variables, say, y, and solve the resulting equation for x(t). If the system is degenerating, stop! A separate analysis is required to determine whether or not there are solutions.

(Shortcut) If possible, use the system to derive an equation that involves y(t) but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for x(t) into this equation to get a formula for y(t). The expressions for x(t), and y(t) give the desired general solution.

Eliminate x from the system and solve for y(t). [Solving for y(t) gives more constants----in fact, twice as many as needed.]

Remove the extra constants by substituting the expressions for x(t) and y(t) into one or both of the equations in the system. Write the expressions for x(t) and y(t) in terms of the remaining constants.

Step 2: Evaluate the given equation

Given that:

$D [x] + (D + 1) [y] = e^{t}$ …… (1)

$D^{2} [x] + (D^{2} + D) [y] = 0$ …… (2)

Multiply D on equation (1).

$D^{2} [x] + D (D + 1) [y] = D [e^{t}] D^{2} [x] + (D^{2} + D) [y] = e^{t} \dots (3)$

Now, compare the equations (3) and (2).

$e^{t} = 0$

Therefore, this condition cannot possible. So, the given system is degenerate.

Most popular questions for Math Textbooks

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).

$\frac{d x}{d t} = x (7 - x - 2 y), \frac{d y}{d t} = y (5 - x - y)$

In Problems 3–6, find the critical point set for the given system.

$\frac{d x}{d t} = x^{2} - 2 x y, \frac{d y}{d t} = 3 x y - y^{2}$

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

$3 x'' + 5 x - 2 y = 0; x (0) = - 1, x' (0) = 0 4 y'' + 2 y - 6 x = 0; y (0) = 1, y' (0) = 2$

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

$\frac{dx}{dt} = x - 4 y \frac{dy}{dt} = x + y$

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

$x'' + 5 x - 4 y = 0, - x + y'' + 2 y = 0$

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free