Problem 13 - Advanced Engineering Mathematics

Question

Find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}^{-1}\left\\{\frac{1}{s^{2}-6 s+10}\right\\} $$

Step-by-Step Solution

Answer

The inverse Laplace transform is \(e^{3t}\sin(t)\).

Step 1: Identify the Form

The expression to be inverted is \(\frac{1}{s^2-6s+10}\). Notice that the denominator is a quadratic expression. We need to rewrite it in a different form to perform the inverse Laplace transform.

Step 2: Complete the Square

Start by completing the square for the quadratic expression in the denominator:\[s^2 - 6s + 10 = (s^2 - 6s + 9) + 1 = (s - 3)^2 + 1.\]The expression \(\frac{1}{s^2-6s+10}\) becomes \(\frac{1}{(s-3)^2+1}\).

Step 3: Recognize the transform

The expression \(\frac{1}{(s-a)^2 + b^2}\) is a standard form in Laplace transforms, where the inverse is known to be:\[f(t) = e^{at} \sin(bt).\]Identify \(a = 3\) and \(b = 1\).

Step 4: Apply the Inverse Transform

Using the formula \(e^{at}\sin(bt)\), we substitute \(a = 3\) and \(b = 1\) to get:\[f(t) = e^{3t} \sin(t).\]Thus, the inverse Laplace transform is \(e^{3t}\sin(t)\).

Key Concepts

Completing the Square

Completing the square is a useful technique in algebra that transforms a quadratic expression into a perfect square trinomial plus a constant. This process not only simplifies expressions but also makes it easier to solve and analyze quadratic equations.
The method involves three main steps:

• Identify the quadratic expression of the form \( ax^2 + bx + c \).
• Take half of the coefficient of \(x\), square it, and then add and subtract this value within the expression.
• Rewrite the quadratic as a perfect square trinomial plus or minus a constant.

For example, consider the expression \( s^2 - 6s + 10 \). To complete the square:

• Take half of \(-6\), which is \(-3\), and square it to get \(9\).
• Rewrite the expression as \( s^2 - 6s + 9 + 1 \).
• Express it as \((s - 3)^2 + 1\).

Completing the square transforms the quadratic expression into a form that is easier to manage, especially when performing inverse Laplace transforms.

Quadratic Expression

A quadratic expression is a polynomial of degree two, represented generally as \( ax^2 + bx + c \). It is a key element in many areas of algebra and is crucial in solving equations, graphing parabolas, and calculus.
Here are the essential components of a quadratic expression:

• The **quadratic term**: \(ax^2\), where \(a\) is a constant that affects the curvature of the graph.
• The **linear term**: \(bx\), which influences the slope or tilt of the parabola on a graph.
• The **constant term**: \(c\), which can shift the graph vertically.

In the problem, the given expression \(s^2 - 6s + 10\) is a quadratic expression. To apply transformations or find the inverse Laplace transform, such expressions need to be brought into a more manageable form, like completing the square. Understanding these expressions helps in analyzing their behavior and solving for unknowns.

Standard Form

Standard form in mathematics often refers to a conventional way of writing equations or expressions that are consistent and easy to work with. In the context of Laplace transforms, certain expressions have standard inverses that are well-documented formulas.
For inverse Laplace transforms:

• The standard form \( \frac{1}{(s-a)^2 + b^2} \) directly corresponds to the inverse \( f(t) = e^{at} \sin(bt) \).
• The parameters \(a\) and \(b\) from this form become part of the exponential and sine functions, respectively, in the inverse.

In the exercise, once we completed the square for \( s^2 - 6s + 10 \) to get \( (s - 3)^2 + 1 \), it fit into this standard form. Identifying \(a = 3\) and \(b = 1\), we were able to utilize the known result for the inverse transform, resulting in \( f(t) = e^{3t} \sin(t) \). Understanding and identifying standard forms is critical as it dramatically simplifies the process of finding inverse Laplace transforms.