Section 11.1 Overview
At this point, the
roadmap introduced at the beginning of this chapter should start to come together. The steps have been explored in previous sections, and now it is time to apply them in sequence to solve a complete initial-value problem. We start by applying the forward Laplace transform to the entire differential equation (Step 1), which temporarily encases the unknown inside
\(Y(s)\text{.}\) Next, you use algebra to isolate
\(Y(s)\) (Step 2a). After that, you prepare for the backward transform by breaking
\(Y(s)\) into a sum of functions that correspond to known Laplace transforms (Step 2b). Finally, you apply the backward transform to
\(Y(s)\) to recover the solution,
\(y(t)\) (Step 3).
The complete strategy for this method is outlined as follows:
Method 6. Laplace Transform Method.
The particular solution to the initial-valued differential equation
\begin{equation}
y^{(n)} + a_{n-1}\, y^{(n-1)} + \cdots + a_1\, y' + a_0\, y = g(t)\tag{47}
\end{equation}
and given initial conditions, follows from these steps:
- Step 1: Apply the Forward Transform
Apply the Laplace transform,
\(\laplacesym\text{,}\) to both sides of
(47). Use the
table and
properties to convert each term into a function of
\(s\text{.}\)
- Step 2a: Solve for \(Y(s)\)
Use algebra to isolate \(Y(s)\) as a function of \(s\text{.}\)
- Step 2b: Prepare for the Backward Transform
Use techniques such as completing the square or partial fraction decompoition to rewrite
\(Y(s)\) as a sum of
\(s\)-functions found in the
table of common transforms.
- Step 3: Apply the Backward Transform
Apply the inverse Laplace transform, \(\laplacesym^{-1}\text{,}\) to both sides of the \(Y(s)\) equation to get the solution \(y(t)\text{.}\)
By following these steps, the Laplace Transform Method provides a clear path from the original differential equation to its solution, leveraging the power of both forward and backward transforms.
Reading Questions Check-Point Questions
The following questions refer to the steps of the Laplace Transform Method discussed above. Assume the dependent variable is \(y(t)\text{.}\)
1. Which of the following best summarizes the Laplace Transform Method?
Which of the following best summarizes the Laplace Transform Method?
- Apply the forward transform, solve for \(Y(s) \text{,}\) prepare for the backward transform, and apply the inverse transform to find \(y(t) \text{.}\)
Correct! The Laplace Transform Method involves applying the forward transform, solving for \(Y(s) \text{,}\) preparing for the backward transform, and applying the inverse transform to find \(y(t) \text{.}\)
- Differentiate the equation, apply the forward transform, solve for \(Y(s) \text{,}\) and find the inverse.
Not quite. The Laplace Transform Method does not involve differentiating the equation but rather applying the forward transform to convert it into an algebraic form.
- Solve the differential equation directly, then use the Laplace transform to verify the solution.
Incorrect. The Laplace Transform Method involves applying the forward transform to simplify the differential equation, not to verify the solution.
- Use algebra to find the solution, then apply the Laplace transform to check the solution.
Incorrect. The Laplace Transform Method involves applying the forward transform to simplify the differential equation, not to check the solution.
2. Match the steps of Laplace Transform Method.
3. After isolating \(Y(s) \text{,}\) which step follows in the Laplace Transform Method?
After isolating \(Y(s) \text{,}\) which step follows in the Laplace Transform Method?
- Apply the backward transform directly.
Not quite. After isolating \(Y(s) \text{,}\) you prepare it for the backward transform by rewriting it as a sum of known Laplace transforms.
- Use integration to find the solution.
Incorrect. After isolating \(Y(s) \text{,}\) you prepare it for the backward transform by rewriting it as a sum of known Laplace transforms.
- Prepare for the backward transform by rewriting \(Y(s) \text{.}\)
Correct! After isolating \(Y(s) \text{,}\) you prepare it for the backward transform by rewriting it as a sum of known Laplace transforms.
- Check the initial conditions.
Not quite. Checking the initial conditions is done in Step 1.
4. Which of the following best describes the purpose of Step 3 in the Laplace Transform Method?
What is the purpose of Step 3 in the Laplace Transform Method?
- To solve for \(Y(s)\) using algebra.
Incorrect. Step 2a involves solving for \(Y(s)\) using algebra.
- To rewrite the equation as a sum of \(s\)-functions.
Incorrect. Step 2b involves rewriting \(Y(s)\) as a sum of known Laplace transforms.
- To recover the solution, \(y(t)\text{,}\) inside \(Y(s)\text{.}\)
Correct! Step 3 involves applying the inverse Laplace transform to recover the solution \(y(t)\text{.}\)
- To apply the forward Laplace transform to \(y(t)\text{.}\)
Incorrect. The forward Laplace transform is applied in Step 1.
5. What is the primary goal of applying the forward Laplace transform to a differential equation?
What is the primary goal of applying the forward Laplace transform to a differential equation?
- To find the solution \(y(t) \) directly.
Not quite. The forward Laplace transform is used to convert the differential equation into an algebraic equation, not to find the solution directly.
- To convert the differential equation into an algebraic equation involving \(Y(s) \text{.}\)
Correct! The forward Laplace transform simplifies the differential equation by converting it into an algebraic form in terms of \(Y(s) \text{.}\)
- To rewrite the differential equation as a sum of functions.
Close, but the main purpose of the forward transform is to convert the differential equation into an algebraic equation involving \(s \text{,}\) not just to rewrite it as a sum of functions.
- To simplify the coefficients of the differential equation.
Incorrect. The Laplace transform does not directly simplify the coefficients but rather transforms the entire differential equation into an algebraic equation in terms of \(s \text{.}\)
6. What is the primary goal of Step 2a in the Laplace Transform Method?
What is the primary goal of Step 2a in the Laplace Transform Method?
- To isolate \(Y(s) \) as a function of \(s \)
Correct! Step 2a involves isolating \(Y(s) \) on one side of the equation.
- To apply the inverse Laplace transform
Incorrect. Applying the inverse Laplace transform is done in Step 3.
- To account for the initial conditions
Incorrect. The initial conditions are accounted for in Step 1.
- To rewrite \(Y(s) \) as a sum of known transforms
Incorrect. Rewriting \(Y(s) \) is done in Step 2b.
7. Match the Next Step in the Method.
Drag the task, on the left, that directly follows the task on the right.- ... The result is your solution.
- After applying the inverse laplace transform to \(Y(s)\) ...
- ... Use algebra to isolate it.
- After transforming the differential equation to introduce \(Y(s)\) ...
- ... Rewrite it as a sum of expressions resembling common Laplace transforms.
- Once you have isolated \(Y(s)\text{...}\)
- ... Apply the Laplace Transform to it
- Given an initial-valued problem ...
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