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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 7: 15E (page 350)

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

Short Answer

Expert verified

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

Step by step solution

01

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.
$F (s) = {鈭�}_{0}^{鈭�} f (t) e^{- st} t'$

02

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

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Most popular questions from this chapter

In Problems 3-10, determine the Laplace transform of the given function. $t^{2} u (t - 4)$

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.

$w'' + w = u (t - - 2) - - u (t - - 4); w (0) = 1, w' (0) = 0$

In Problems 1鈥�19, use the method of Laplace transforms to solve the given initial value problem. Here x鈥�, y鈥�, etc., denotes differentiation with respect to t; so does the symbol D.

$\begin{array}{l} D [x] + y = 0; 鈥夆赌夆赌� x (0) = 7 / 4, \\ 4 x + D [y] = 3; 鈥夆赌夆赌� y (0) = 4 \end{array}$

In Problems 1 and 2, use the definition of the Laplace transform to determine $L {f}$ .

$f (t) = {\begin{cases} e^{- t}, 0 鈮� t 鈮� 5 \\ - 1, 5 < t \end{cases}$

In Problems $1 - 14$ , solve the given initial value problem using the method of Laplace transforms.

$3 . y'' + 6 y' + 9 y = 0; y (0) = - 1, y' (0) = 6$

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