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Exploring Differential Equations An Interactive, Student-Centered Approach

Geoffrey W. Cox, Ph.D.

Section 11.5 The Laplace Transform Reference Guide

In this final section, we summarize everything we’ve done. Below, you will find tables for the most common Laplace transforms and the properties used to combine and manipulate them. These will provide a useful resource in the next chapter, where we begin solving differential equations using the Laplace transform method.
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Subsection Laplace Transform Tables

We have done a lot of work computing common transforms and deriving the properties of the Laplace transform. Now, we can summarize all of this information in tables for quick reference.
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We begin with the specific transforms of exponentials, polynomial terms, sine/cosine, and the products of these functions. In later chapters, we will encounter new types of functions that we need to transform, so this table will continue to grow as we go.
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Table 192. Table of Common Laplace Transforms
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Function
(\(t\)-Domain)
\begin{equation*} f(t) \end{equation*}
Laplace Transform
(\(s\)-Domain)
\begin{equation*} \lap{f(t)} = F(s) \end{equation*}

Existence
Condition
L\(_1\)
\begin{equation*} 1 \end{equation*}
\begin{equation*} \frac{1}{s} \end{equation*}
\begin{equation*} s > 0 \end{equation*}
L\(_2\)
\begin{equation*} e^{at} \end{equation*}
\begin{equation*} \frac{1}{s - a} \end{equation*}
\begin{equation*} s > a \end{equation*}
L\(_3\)
\begin{equation*} t^n \end{equation*}
\begin{equation*} \frac{n!}{s^{n+1}} \end{equation*}
\begin{equation*} s > 0 \end{equation*}
L\(_4\)
\begin{equation*} \sin(bt) \end{equation*}
\begin{equation*} \frac{b}{s^2+b^2} \end{equation*}
\begin{equation*} s > 0 \end{equation*}
L\(_5\)
\begin{equation*} \cos(bt) \end{equation*}
\begin{equation*} \frac{s}{s^2+b^2} \end{equation*}
\begin{equation*} s > 0 \end{equation*}
L\(_6\)
\begin{equation*} e^{at}\ t^n \end{equation*}
\begin{equation*} \frac{n!}{(s-a)^{n+1}} \end{equation*}
\begin{equation*} s > a \end{equation*}
L\(_7\)
\begin{equation*} e^{at}\sin(bt) \end{equation*}
\begin{equation*} \frac{b}{(s-a)^2+b^2} \end{equation*}
\begin{equation*} s > a \end{equation*}
L\(_8\)
\begin{equation*} e^{at}\cos(bt) \end{equation*}
\begin{equation*} \frac{s-a}{(s-a)^2+b^2} \end{equation*}
\begin{equation*} s > a \end{equation*}
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Checkpoint 193. 📖❓ What’s Missing in each Transform?

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The next set of transforms are more general rules you can apply in different situations.
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Table 194. Table of Laplace Transform Rules
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Function (\(t\)-Domain) Laplace Transform (\(s\)-Domain) Existence
\begin{equation*} f(t) \end{equation*}
\begin{equation*} \lap{f(t)} = F(s) \end{equation*}
Condition
R\(_{1}\)
\begin{equation*} f'(t) \end{equation*}
\begin{equation*} sF(s) - f(0) \end{equation*}
\begin{equation*} s > 0 \end{equation*}
R\(_{2}\)
\begin{equation*} f''(t) \end{equation*}
\begin{equation*} s^2F(s) - sf(0) - f'(0) \end{equation*}
\begin{equation*} s > 0 \end{equation*}
R\(_{3}\)
\begin{equation*} f'''(t) \end{equation*}
\begin{equation*} s^3F(s) - s^2f(0) - sf'(0) - f''(0) \end{equation*}
\begin{equation*} s > 0 \end{equation*}
R\(_{4}\)
\begin{equation*} e^{at} f(t) \end{equation*}
\begin{equation*} F(s-a) \end{equation*}
\begin{equation*} s > a \end{equation*}
R\(_{5}\)
\begin{equation*} t^n f(t) \end{equation*}
\begin{equation*} (-1)^n \frac{d^n}{ds^n}\Big[F(s)\Big] \end{equation*}
\begin{equation*} s > 0 \end{equation*}
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Finally, we list the Laplace transform properties. It is short for now, but we will add more to this table later.
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Table 195. Table of Laplace Transform Properties
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Property Type Property
P\(_{1}\) Linearity
\begin{equation*} \lap{ a f(t) \pm b g(t) } = a \lap{f(t)} \pm b \lap{g(t)} \end{equation*}
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Subsection Next Step: Solving Differential Equations

Everything you’ve learned in this chapter prepares you to solve differential equations using the Laplace transform. In the next chapter, you will see how these tools turn the process of solving differential equations into solving simple algebraic equations, especially when initial conditions are involved.
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Checkpoint 196. 🤔💭 Reading Questions.

(a) 🤔💭 Transform of \(\sin(5t)\).
\(\ds\lap{\sin(5t)} = \) ?
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  • \(\ds\frac{5}{s^2 + 25}\)
  • Correct! This follows the formula \(\lap{\sin(bt)} = \frac{b}{s^2 + b^2}\) with \(b = 5\text{.}\)
  • \(\ds\frac{s}{s^2 + 25}\)
  • Incorrect. That’s the transform of \(\cos(5t)\text{,}\) not \(\sin(5t)\text{.}\)
  • \(\ds\frac{1}{s^2 + 25}\)
  • Incorrect. The frequency coefficient \(5\) must appear in the numerator.
  • \(\ds\frac{5}{s^2 + 5^2}\)
  • This is mathematically correct but not simplified. Choose the version with \(25\) in the denominator.
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(b) 🤔💭 Transform of \(\cos(-3t)\).
\(\ds\lap{\cos(-3t)} = \) ?
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  • \(\ds\frac{3}{s^2 + 9}\)
  • Incorrect. The numerator for cosine should be \(s\text{,}\) not the frequency.
  • \(\ds\frac{s}{s^2 - 9}\)
  • Incorrect. The denominator should be \(s^2 + 9\text{,}\) not \(s^2 - 9\text{.}\)
  • \(\ds\frac{s}{s^2 + 9}\)
  • Correct! Cosine is even, so \(\cos(-3t)\) = \(\cos(3t)\text{,}\) and its transform is \(\ds\frac{s}{s^2 + 9}\text{.}\)
  • \(\ds\frac{3}{s^2 - 9}\)
  • Incorrect. Both the numerator and denominator are wrong for cosine transforms.
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