Problem 48 - Advanced Engineering Mathematics

Question

Suppose that \(\mathscr{L}\left\\{f_{1}(t)\right\\} \quad F_{1}(s)\) for \(s>c_{1}\) and that \(\mathscr{L}\left\\{f_{2}(t)\right\\} \quad F_{2}(s)\) for \(s>c_{2}\). When does \(\mathscr{L}\left\\{f_{1}(t)+f_{2}(t)\right\\}\) \(F_{1}(s)+F_{2}(s) ?\)

Step-by-Step Solution

Answer

The transform is \( F_{1}(s) + F_{2}(s) \) for \( s > \max(c_{1}, c_{2}) \).

Step 1: Understand the Problem

We are given two Laplace transforms: \( \mathscr{L}\{f_{1}(t)\} = F_{1}(s) \) for \( s > c_{1} \) and \( \mathscr{L}\{f_{2}(t)\} = F_{2}(s) \) for \( s > c_{2} \). We need to find the Laplace transform of the sum \( f_{1}(t) + f_{2}(t) \).

Step 2: Recall the Linearity Property of Laplace Transforms

The linearity property states that the Laplace transform of a sum of functions is the sum of their Laplace transforms, i.e., \( \mathscr{L}\{f_{1}(t) + f_{2}(t)\} = \mathscr{L}\{f_{1}(t)\} + \mathscr{L}\{f_{2}(t)\} \).

Step 3: Apply the Linearity to the Given Functions

Applying linearity, we have \( \mathscr{L}\{f_{1}(t) + f_{2}(t)\} = F_{1}(s) + F_{2}(s) \).

Step 4: Determine the Domain of Validity

Since \( \mathscr{L}\{f_{1}(t)\} = F_{1}(s) \) is valid for \( s > c_{1} \) and \( \mathscr{L}\{f_{2}(t)\} = F_{2}(s) \) is valid for \( s > c_{2} \), the Laplace transform of \( f_{1}(t) + f_{2}(t) \) is valid for \( s > \max(c_{1}, c_{2}) \).

Key Concepts

Linearity Property

The linearity property is a fundamental concept in the realm of Laplace transforms. This property makes the operation particularly useful when dealing with differential equations, especially in engineering and physics applications. When we talk about the linearity property, we are referring to the ability to simplify the transformation of sums of functions.

In essence, if you have two functions, say \( f_1(t) \) and \( f_2(t) \), the Laplace transform of their sum is the sum of their individual Laplace transforms. Mathematically, this is expressed as:
\[ \mathscr{L}\{f_1(t) + f_2(t)\} = \mathscr{L}\{f_1(t)\} + \mathscr{L}\{f_2(t)\} \]

This result is powerful because it allows us to break down complex functions into simpler parts. We then take advantage of already known transforms to find the Laplace transform of the entire expression.

When applying this property, remember:

• Both functions must be piecewise continuous and of exponential order.
• The sum \( f_1(t) + f_2(t) \) should also meet the conditions for the existence of a Laplace transform.

Domain of Validity

The domain of validity is crucial when operating with Laplace transforms. It ensures the results we calculate using transforms make sense in an applied context. Simply put, it's about determining when and where the Laplace transform exists for a given function.

For our functions \( f_1(t) \) and \( f_2(t) \), their respective Laplace transforms are denoted as \( F_1(s) \) and \( F_2(s) \). Each of these is valid in different domains: \( s > c_1 \) and \( s > c_2 \) respectively. The domain of validity for the sum \( f_1(t) + f_2(t) \) thus must satisfy the more stringent of these conditions.

The combined transform \( \mathscr{L}\{f_1(t) + f_2(t)\} = F_1(s) + F_2(s) \) holds true for:
\[ s > \max(c_1, c_2) \]

It's critical to identify the largest valid domain for both functions, ensuring that their sum behaves well in this specified range.

To summarize:

• The maximum requirement \( s > \max(c_1, c_2) \) ensures consistency and accuracy.
• This step is necessary to avoid invalid results that occur outside accepted boundaries.

Functions Sum

When considering the sum of functions in the context of Laplace transforms, it's important to understand how the operations affect both individual functions and their sum. The sum of functions introduces complexity, but thanks to linearity, we can manage it efficiently.

In the scenario of the functions \( f_1(t) \) and \( f_2(t) \), examining them individually makes it easier to apply their transforms and then combine them according to the linearity property. This advantage reduces potential errors when dealing with more complicated expressions that might be intimidating if tackled as a whole.

Here's the basic rundown:

• Analyze each function separately to confirm they meet transformation conditions.
• Apply the linearity principle, then sum their Laplace transforms.
• Ensure the unified result respects the domain of validity, \( s > \max(c_1, c_2) \).

Remember that understanding the step-by-step process of breaking down complex terms into simpler components, applying known transforms, and ensuring the sum lies within a valid domain, is essential for mastering Laplace transforms.