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15E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 350

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.
$y'' - 3 y' + 2 y = cost; y (0) = 0, y' (0) = - 1$

The Initial value for $y'' - 3 y' + 2 y = \cos t$ is $Y (s) = \frac{- s^{2} + s - 1}{(s^{2} + 1) (s - 1) (s - 2)}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the Laplace Transform

The Laplace transform is a strong integral transform used in mathematics to convert a function from the time domain to the s-domain.
In some circumstances, the Laplace transform can be utilized to solve linear differential equations with given beginning conditions.
$F (s) = \int_{0}^{\infty} f (t) e^{- st} t'$

Step 2: Determine the Laplace transform

Applying the Laplace transform and using its linearity as follows:

$L \{y'' - 3 y' + 2 y\} = L \{\cos t\} L \{y''\} - 3 L \{y'\} + 2 L \{y\} = \frac{s}{s^{2} + 1}$

Solve for the transfer function as:

$[s^{2} Y (s) - s y (0) - y' (0)] - 3 [s Y (s) - y (0)] + 2 Y (s) = \frac{s}{s^{2} + 1} s^{2} Y (s) + 1 - 3 s Y (s) + 2 Y (s) = \frac{s}{s^{2} + 1} (s^{2} - 3 s + 2) Y (s) = \frac{s}{s^{2} + 1} - 1 Y (s) = \frac{- s^{2} + s - 1}{(s^{2} - 3 s + 2) (s^{2} + 1)}$

Since $s^{2} - 3 s + 2 = (s - 1) (s - 2)$

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

Therefore, the Initial value for $y'' - 3 y' + 2 y = \cos t$ is $Y (s) = \frac{- s^{2} + s - 1}{(s^{2} + 1) (s - 1) (s - 2)}$

Most popular questions for Math Textbooks

Find the Laplace transform of . $f (t) = \int_{0}^{t} e^{v} s i n (t - v) d v$

Solve the given integral equation or integro-differential equation for y(t) $y (t) + 3 \int_{0}^{t} y (v) s i n (t - v) d v = t$

Determine the inverse Laplace transform of the given function.

$\frac{1}{s^{2} + 4 s + 8}$ .

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$3 t^{2} - e^{2 t}$

In problem 15-22, solve the given integral equation or integro-differential equation for $y (t)$

$y (t) + L \int_{0}^{t} (t - v) y (v) d v = t^{2}$

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