Problem 1 - Advanced Engineering Mathematics

Question

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t e^{10 t}\right\\} $$

Step-by-Step Solution

Answer

The Laplace transform is \( \frac{1}{(s-10)^2} \), assuming \( \Re(s) > 10 \).

Step 1: Identify the Laplace Transform Formula

To find the Laplace transform \(F(s)\) of a given function, we use the formula: \[ \mathscr{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt. \] In this problem, \(f(t) = t e^{10t}\), so we'll apply this formula to \(t e^{10t}\).

Step 2: Write the Integral for the Laplace Transform

The Laplace transform of \(t e^{10t}\) is given by: \[\mathscr{L}\{t e^{10t}\} = \int_{0}^{\infty} e^{-st} t e^{10t} \, dt.\] We can rewrite this integral as: \[\int_{0}^{\infty} t e^{(10-s)t} \, dt.\]

Step 3: Simplify the Integral Using a Known Transform

The integral \(\int_{0}^{\infty} t e^{(a)t} \, dt\) is known to have the solution \(\frac{1}{(s-a)^2}\), applicable when \(\Re(s) > a\). \In the given problem, \(a = 10\), so you'll apply this formula as: \[ \mathscr{L}\{t e^{10t}\} = \frac{1}{(s-10)^2}.\]

Step 4: Define Conditions for the Transform

The condition that \(\Re(s) > a\) must be met for the transform to be valid. \Since \(a = 10\), the condition is \(\Re(s) > 10\). \Thus, the Laplace transform exists and is valid under these conditions.

Key Concepts

Integration Techniques

In the world of calculus, finding the Laplace transform of a function like \(t e^{10t}\) requires adept integration skills. Since the Laplace transform involves the integration of the function multiplied by \(e^{-st}\), knowing the base integration technique is crucial. Here are some basic integration methods to consider:

• **Substitution Method:** Useful when you can relate the integral's variables through substitution.
• **Integration by Parts:** Ideal for products of two functions, like our exercise. This method follows the formula: \( \int u \, dv = uv - \int v \, du \).
• **Definite Integral:** The Laplace transform always uses a definite integral, with limits from 0 to infinity.

It is important to become familiar with these methods and recognize scenarios fitting each technique. In our exercise, integration by parts fits well because we're dealing with a product \(t e^{10t}\). Breaking down the integral into manageable parts simplifies the problem considerably, helping us reach the solution efficiently.

Exponential Functions

Exponential functions form a backbone of many mathematical transformations, including the Laplace transform. An exponential function can be generally expressed as \(f(t) = e^{at}\), where \'a\' determines its growth or decay. In our exercise, the exponential part is \(e^{10t}\):

• **Growth and Decay:** The sign of \(a\) plays a critical role—positive \(a\) signals rapid growth, while negative \(a\) implies decay.
• **Continuous Compounding:** Exponentials are often used to model processes of continuous growth, such as compound interest.
• **Laplace Influence:** In the context of the Laplace transform, exponential functions like \(e^{-st}\) modify the integration process by damping the original function.

When applying the Laplace transform, the exponential \(e^{-st}\) effectively changes how \(f(t)\) behaves under integration, often simplifying otherwise complex calculations. Understanding exponential characteristics aids in predicting the behavior of transformed functions and solving differential equations.

Piecewise Functions

In mathematics, piecewise functions can be thought of as a collection of 'pieces,' each with its own rule in specific intervals. Though the current exercise does not explicitly involve piecewise functions, the concept can help one understand how Laplace transforms handle "switched" functions, especially when dealing with discontinuities.

• **Defining by Intervals:** Each piece is defined by different expressions based on intervals such as \(f(t) = t\) for \(t < a\), and \(f(t) = e^{at}\) for \(t \ge a\).
• **Continuity & Discontinuity:** Key characteristics include how functions behave at the boundaries of these intervals (i.e., whether they are continuous or have jumps).
• **Transform Impact:** In the Laplace domain, piecewise functions are transformed considering each piece separately but integrated over its whole domain.

When assessing transforms, knowing the distinct behavior across intervals lets you effectively use tools like the Heaviside function to simplify handling these transitions. Thus, harnessing the power of piecewise representations can make solving complex transformations straightforward.