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

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.

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

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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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

Step by step solution

Derive the given equation using Laplace transformation

Use initial condition and find the Y variable

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

Short Answer

Step by step solution

Derive the given equation using Laplace transformation

Use initial condition and find the Y variable

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Pure Maths

Probability and Statistics

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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