Section D.3 Orphaned Content

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Additional Narrative.

Additional Examples.

Example D.19. Example 1: Solving a First-Order Linear Differential Equation.

Consider the differential equation:

y^{'} + 3 y = 6 e^{2 t}, y (0) = 1

Applying the Laplace transform to both sides, we get:

s Y (s) - y (0) + 3 Y (s) = \frac{6}{s - 2}

Substituting the initial condition

y (0) = 1,

the equation becomes:

s Y (s) - 1 + 3 Y (s) = \frac{6}{s - 2}

Rearranging and solving for

Y (s) :

Y (s) (s + 3) = 1 + \frac{6}{s - 2}

Y (s) = \frac{1}{s + 3} + \frac{6}{(s - 2) (s + 3)}

We can now decompose the second term using partial fractions:

\frac{6}{(s - 2) (s + 3)} = \frac{A}{s - 2} + \frac{B}{s + 3}

Solving for

A

and

B,

we get

A = 2

and

B = 4 .

Therefore:

Y (s) = \frac{1}{s + 3} + \frac{2}{s - 2} + \frac{4}{s + 3}

Y (s) = \frac{5}{s + 3} + \frac{2}{s - 2}

Taking the inverse Laplace transform, we obtain the solution:

y (t) = 5 e^{- 3 t} + 2 e^{2 t}

Example D.20. Example 2: Second-Order Differential Equation.

Consider the second-order differential equation:

y^{″} + 4 y = \cos (2 t), y (0) = 0, y^{'} (0) = 1

Applying the Laplace transform:

s^{2} Y (s) - s y (0) - y^{'} (0) + 4 Y (s) = \frac{s}{s^{2} + 4}

Substituting the initial conditions:

s^{2} Y (s) - 1 + 4 Y (s) = \frac{s}{s^{2} + 4}

Solving for

Y (s) :

Y (s) (s^{2} + 4) = 1 + \frac{s}{s^{2} + 4}

After rearranging and solving, the inverse Laplace transform gives:

y (t) = \sin (2 t) + t

Reading Questions Additional Practice

1. Which of the following is NOT a technique mentioned for preparing $Y (s)$ for the backward transform?

Which of the following is NOT a technique mentioned for preparing $Y (s)$ for the backward transform?

Completing the square
Incorrect. Completing the square is a technique used to rewrite $Y (s)$ as a sum of known Laplace transforms.
Partial fraction decomposition
Incorrect. Partial fraction decomposition is another technique used to prepare $Y (s)$ for the inverse transform.
Integration by Parts
Correct! Integration by parts is not a technique used to prepare $Y (s)$ for the backward transform.
Rewriting as a sum of $s$ -functions
Incorrect. Rewriting $Y (s)$ is a technique used in Step 2b.

2. What is the goal of Step 2b in the Laplace Transform Method?

What is the goal of Step 2b in the Laplace Transform Method?

To apply the forward Laplace transform
Incorrect. Applying the forward transform is done in Step 1.
To solve for $y (t)$
Incorrect. Solving for $y (t)$ is not the goal of Step 2b.
To rewrite $Y (s)$ as a sum of $s$ -functions found in the table of common transforms
Correct! Step 2b involves breaking down $Y (s)$ into simpler components that match known Laplace transforms.
To find the particular solution to the differential equation
Incorrect. Finding the particular solution is the ultimate goal of the Laplace Transform Method, not just Step 2b.

3. After applying the backward Laplace transform in Step 3, you obtain $y (t),$ the $\underset{―}{}$ to the differential equation.

After applying the backward Laplace transform in Step 3, you obtain $y (t),$ the to the differential equation

4. Which of the following is the main purpose of Step 2b in the Laplace Transform Method?

Which of the following is the main purpose of Step 2b in the Laplace Transform Method?

To solve the algebraic equation for $Y (s)$
Incorrect. Solving for $Y (s)$ happens in Step 2a.
To prepare $Y (s)$ for the inverse Laplace transform by rewriting it as a sum of known forms
Correct! Step 2b involves breaking down $Y (s)$ into simpler components that match known Laplace transforms.
To apply the forward Laplace transform
Incorrect. Applying the forward transform happens in Step 1.
To account for initial conditions
Incorrect. Initial conditions are accounted for in Step 1.

5. Which of the statements are true?

Which of the statements are true?

The forward Laplace transform converts a differential equation into an algebraic equation
This statement is true. The forward Laplace transform simplifies the differential equation by converting it into an algebraic equation in terms of $Y (s) .$
In Step 2a, you isolate $y (t) .$
This statement is false. In Step 2a, you solve for $Y (s),$ not $y (t) .$
Step 2a involves applying the backward Laplace transform
This statement is false. Step 2a involves solving for $Y (s)$ as a function of $s .$
Step 2b involves rewriting $Y (s)$ to match forms in the common Laplace transform table
This statement is true. Step 2b prepares $Y (s)$ for the backward transform by breaking it into known forms found in the table of common Laplace transforms.
The final step involves applying the inverse Laplace transform to recover the solution $y (t)$
This statement is true. The final step of the Laplace Transform Method involves applying the inverse Laplace transform to recover the solution $y (t) .$

6. Which step is likely to involve completing the square?

Which step is likely to involve completing the square?

Step 1: Apply the Forward Transform
Incorrect. Completing the square is not part of applying the forward transform.
Step 2a: Solve for $Y (s)$
Incorrect. Completing the square typically happens in Step 2b.
Step 2b: Prepare for the Backward Transform
Correct! Completing the square is a technique used in Step 2b to simplify $Y (s) .$
Step 3: Apply the Backward Transform
Incorrect. Completing the square should be done before applying the backward transform.

7. Similar to other methods, this method applies the initial conditions to the general solution to find a particular solution.

Similar to other methods, this method applies the initial conditions to the general solution to find a particular solution

True
Incorrect.
False
Correct! The Laplace Transform Method accounts for initial conditions in Step 1.

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