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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: 21E (page 350)

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

Short Answer

Expert verified

The solution for the Laplace transformation is

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

Step by step solution

01

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.

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

02

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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Most popular questions from this chapter

Determine the inverse Laplace transform of the given function.

$\frac{e^{- 2 s} (4 s + 2)}{(s - 1) (s + 2)}$

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

$y'' + 4 y = g (t); y (0) = 1, y' (0) = 3, w h e r e g (t) = {\begin{cases} s i n t, 0 猢� t 猢� 2 蟺, \\ 0, 2 蟺 < t \end{cases}$

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.]

${te}^{2 t} \cos 5 t$

In Problems 3鈥�10, determine the Laplace transform of the given function. $e^{3 t} s i n 4 t$

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.]

$2 t^{2} e^{- t} - t + \cos 4 t$ .

See all solutions

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