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.

Short Answer

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.
$x' = 3 x - 2 y + sint, y' = 4 x - y - cost$

The solutions for the given linear system are;

$x (t) = c_{1} e^{t} \cos 2 t + c_{2} e^{t} \sin 2 t + \frac{7}{10} cost - \frac{1}{10} sint$ and

$y (t) = (c_{1} - c_{2}) e^{t} \cos 2 t + (c_{1} + c_{2}) e^{t} \sin 2 t + \frac{11}{10} cost + \frac{7}{10} sint$ .

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.

The differential for only x(t) is $(L_{1} \times L_{4} - L_{2} \times L_{3}) [x] = L_{4} [g_{1} (t)] - L_{1} [g_{2} (t)]$ .

Step 2: Evaluate the given equation

Given that,

$x' = 3 x - 2 y + sint \dots (1)$

$y' = 4 x - y - cost \dots (2)$

Let us rewrite this system of operators in operator form:

$(D - 3) [x] + 2 y = sint \dots (3)$

$- 4 x + (D + 1) [y] = - cost \dots (4)$

Multiply (D+1) on both sides of equation (3) and multiply -2 on both sides of equation (4) then add the founded equations together.

$(D + 1) (D - 3) [x] + 2 (D + 1) y = (D + 1) [sint] 8 x - 2 (D + 1) [y] = 2 cost (D^{2} - 3 D + D - 3) [x] + 8 x = (D + 1) [sint] + 2 cost$

$(D^{2} - 2 D - 3 + 8) [x] = cost + sint + 2 cost (D^{2} - 2 D + 5) [x] = 3 cost + sint$

$(D^{2} - 2 D + 5) [x] = 3 cost + sint \dots (5)$

Since the auxiliary equation to the corresponding homogeneous equation is $r^{2} - 2 r + 5 = 0$ . The roots are $r = 1 + 2 i$ and $r = 1 - 2 i$ .

Then, the homogeneous solution of x is $x_{h} (t) = c_{1} e^{t} \cos 2 t + c_{2} e^{t} \sin 2 t \dots (6)$

Let us take the undetermined coefficients and assume that;

$x_{p} (t) = Acost + Bsint \dots (7)$

Now derivate the equation (7)

$D [x_{p} (t)] = - Asint + Bcost D^{2} [x_{p} (t)] = - Acost - Bsint$

Step 3: Substitution method

Substitute the derivative in equation (5).

$(D^{2} - 2 D + 5) [x_{p}] = 3 cost + sint D^{2} [x_{p}] - 2 D [x_{p}] + 5 [x_{p}] = 3 cost + sint - Acost - Bsint + 2 Asint - 2 Bcost + 5 Acost + 5 Bsint = 3 cost + sint 4 Acost + 2 Asint + 4 Bsint - 2 Bcost = 3 cost + sint (4 A - 2 B) cost + (4 B + 2 A) sint = 3 cost + sint$

Now, equalize the like terms.

$4 A - 2 B = 3 4 B + 2 A = 1$

Solve the equations to find the value of A and B.

$4 A - 2 B = 3 4 A + 8 B = 2 - - - - 10 B = 1 B = - \frac{1}{10}$

Then,

$4 A - 2 (- \frac{1}{10}) = 3 4 A + \frac{1}{5} = 3 20 A + 1 = 15 20 A = 14 A = \frac{14}{20} = \frac{7}{10}$

So, $x_{p} (t) = \frac{7}{10} cost - \frac{1}{10} sint \dots (8)$

Use equations (6) and (8) to get,

$x (t) = x_{h} (t) + x_{p} (t)$

$x (t) = c_{1} e^{t} \cos 2 t + c_{2} e^{t} \sin 2 t + \frac{7}{10} cost - \frac{1}{10} sint \dots (9)$

Step 4: Substitution method

Substitute the equation (9) in equation (3).

$2 y = (3 - D) [x] + sint 2 y = (3 - D) [c_{1} e^{t} \cos 2 t + c_{2} e^{t} \sin 2 t + \frac{7}{10} cost - \frac{1}{10} sint] + sint 2 y = 3 c_{1} e^{t} \cos 2 t + 3 c_{2} e^{t} \sin 2 t + \frac{21}{10} cost - \frac{3}{10} sint - D [c_{1} e^{t} \cos 2 t + c_{2} e^{t} \sin 2 t + \frac{7}{10} cost - \frac{1}{10} sint] + sint 2 y = 3 c_{1} e^{t} \cos 2 t + 3 c_{2} e^{t} \sin 2 t + \frac{21}{10} cost - \frac{3}{10} sint - [c_{1} e^{t} \cos 2 t - 2 c_{1} e^{t} \sin 2 t + c_{2} e^{t} \sin 2 t + 2 c_{2} e^{t} \cos 2 t - \frac{7}{10} sint - \frac{1}{10} cost] + sint 2 y = (2 c_{1} - 2 c_{2}) e^{t} \cos 2 t + (2 c_{1} + 2 c_{2}) e^{t} \sin 2 t + \frac{11}{5} cost + \frac{7}{5} sint y (t) = (c_{1} - c_{2}) e^{t} \cos 2 t + (c_{1} + c_{2}) e^{t} \sin 2 t + \frac{11}{10} cost + \frac{7}{10} sint$

Thus, the solutions for the given linear system are;

$x (t) = c_{1} e^{t} \cos 2 t + c_{2} e^{t} \sin 2 t + \frac{7}{10} cost - \frac{1}{10} sint$ and $y (t) = (c_{1} - c_{2}) e^{t} \cos 2 t + (c_{1} + c_{2}) e^{t} \sin 2 t + \frac{11}{10} cost + \frac{7}{10} sint$

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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.
$x' = 3 x - 2 y + sint, y' = 4 x - y - cost$

Step by Step Solution

Step 1: General form

Step 2: Evaluate the given equation

Step 3: Substitution method

Step 4: Substitution method

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

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'=3x-2y+sint,y'=4x-y-cost

Step by Step Solution

Step 1: General form

Step 2: Evaluate the given equation

Step 3: Substitution method

Step 4: Substitution method

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

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.
$x' = 3 x - 2 y + sint, y' = 4 x - y - cost$