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 $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' - 2 y' + y = c o s t - s i n t; y (0) = 1, y' (0) = 3$

The solution for the Laplace transformation is

$Y = \frac{s^{3} + s^{2} + 2 s}{(s^{2} + 1) {(s - 1)}^{2}}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Derive the given equation using Laplace transformation

Define $L \{y\} (s) = Y (s)$

$L \{y''\} - 2 L \{y'\} + L \{y\} = L \{\cos (t)\} - L \{\sin (t)\}$

Using the properties listed below; take the Laplace transform of the equation.

$L \{y'\} (s) = sL \{y\} (s) - y (0) L \{y''\} (s) = s^{2} L \{y\} (s) - sy (0) - y' (0) L \{\cos (bt)\} = \frac{s}{s^{2} + b^{2}} L \{\sin (bt)\} = \frac{b}{s^{2} + b^{2}}$

Substitute the properties into the equation.

$[s^{2} Y - s y (0) - y' (0)] - 2 [s Y - y (0)] + Y = \frac{s}{s^{2} + 1} - \frac{1}{s^{2} + 1}$

Step 2: Use initial condition and find the Y variable

Solve for the Laplace transform as:

$s^{2} Y - s - 3 - 2 s Y + 2 + Y = \frac{s - 1}{s^{2} + 1}$

Substitute the initial conditions:

$y (0) = 1 a n d y^{'} (0) = 3$

Isolate the Y variable.

$s^{2} Y - 2 s Y + Y - s - 1 = \frac{s - 1}{s^{2} + 1} Y (s^{2} - 2 s + 1) = \frac{s - 1}{s^{2} + 1} + s + 1 Y (s - 1)^{2} = \frac{s - 1 + (s + 1) (s^{2} + 1)}{s^{2} + 1} Y = \frac{s^{3} + s^{2} + 2 s}{(s^{2} + 1) {(s - 1)}^{2}}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' - 2 y' + y = c o s t - s i n t; y (0) = 1, y' (0) = 3$

Step by Step Solution

Step 1: Derive the given equation using Laplace transformation

Step 2: Use initial condition and find the Y variable

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 15-24, solve for Y(s), the Laplace transform of the solution yt to the given initial value problem.y''-2y'+y=cost-sint; y0=1, y'0=3

Step by Step Solution

Step 1: Derive the given equation using Laplace transformation

Step 2: Use initial condition and find the Y variable

Most popular questions for Math Textbooks

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In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' - 2 y' + y = c o s t - s i n t; y (0) = 1, y' (0) = 3$