Problem 8 - Advanced Engineering Mathematics

Question

Fill in the blanks or answer true/false. $$ \mathscr{L}\left\\{t e^{-7 t}\right\\}= $$____

Step-by-Step Solution

Answer

\( \frac{1}{(s+7)^2} \)

Step 1: Understand the Given Function

We need to find the Laplace transform of the function \( f(t) = t e^{-7t} \). The Laplace transform \( \mathscr{L}\{f(t)\} \) is a technique used to convert differential equations into algebraic equations.

Step 2: Recall the Laplace Transform Formula

The Laplace transform of a function \( t^n e^{-at} \) is given by: \[ \mathscr{L}\{t^n e^{-at}\} = \frac{n!}{(s+a)^{n+1}} \]In this problem, \( n = 1 \) and \( a = 7 \).

Step 3: Apply the Formula

Substitute \( n = 1 \) and \( a = 7 \) into the formula: \[ \mathscr{L}\{t e^{-7t}\} = \frac{1!}{(s+7)^{1+1}} \]

Step 4: Simplify the Expression

Calculate the factorial and simplify: \[ \mathscr{L}\{t e^{-7t}\} = \frac{1}{(s+7)^2} \]This is the Laplace transform of the given function.

Key Concepts

Differential Equations

Differential equations involve mathematical equations that relate a function with its derivatives. They serve as a foundational component in analyzing systems where the rate of change is important. For instance, they often describe physical phenomena in engineering and natural sciences, such as how heat spreads through a solid object or how populations grow over time.

• A differential equation says how a function's value changes over time.
• It includes derivatives, which are expressions showing the rate of change.
• The general solution to a differential equation involves integration.

By using the Laplace transform, we can take these differential equations and turn them into algebraic equations. This simplifies the solving process by converting the functions into a different domain. Understanding differential equations is key since they appear in various real-world scenarios.

Algebraic Equations

Algebraic equations are equations involving polynomials and other algebraic expressions. They are considered simpler than differential equations because they deal with expressions that might only involve constants, variables, and arithmetic operations.

• Algebraic equations are settled by finding values of variables that satisfy the equation.
• They don't include derivatives, unlike differential equations.

When we use Laplace Transform on a differential equation, the biggest advantage is turning it into an algebraic equation. This shift makes complex problems easier to handle, especially in fields like control systems engineering, where solutions can be readily analyzed.

Transform Techniques

Transform techniques are mathematical strategies utilized to convert functions from one domain to another, aiding in easier problem-solving. The Laplace transform is a primary example—allowing us to move from the time domain, where differential equations live, to the s-domain, where algebraic equations can be manipulated.

• Laplace Transform is a powerful tool due to its ability to handle initial conditions directly.
• The transform is particularly useful when dealing with linear time-invariant systems.
• It simplifies solving integral and differential equations.

By using transform techniques like Laplace Transform, solving complex equations becomes more manageable. Once everything is simplified in the s-domain, inverse transforms are typically used to revert back to the time domain for real-world interpretation. Understanding transform techniques enhances one's ability to tackle numerous mathematical models effectively.