import sympy as sym
from sympy.abc import s,t
a = sym.symbols('a', real=True, nonzero=True)
b = sym.symbols('b', real=True, nonzero=True)
def L(f):
return sym.laplace_transform(f, t, s, noconds=True)4.2: Properties of Laplace Transforms
Reading: Notes on Diffy Q’s Section 6.1 Part A and Section 6.2
Recall in the materials for Worksheet 20 Introduction to Laplace Transforms we used the definition of the Laplace transform,
\[\Large{\color{dodgerblue}{ \mathscr{L} \left\{ f \right\} = F(s) = \int_0^{\infty} e^{-st}f(t) \ dt}},\]
to find the Laplace transform of some common functions. For example, we showed that:
- \(\mathscr{L} \left\{ t \right\} = \dfrac{1}{s^2}\) for \(s > 0\).
- \(\mathscr{L} \left\{ e^{at} \right\} = \dfrac{1}{s-a}\) for \(s > a\).
In this worksheet, we will explore some useful properties that Laplace transforms statisfy that will allow us to use what we know already know about \(\mathscr{L} \left\{ t \right\}\) and \(\mathscr{L} \left\{ e^{at} \right\}\) to evaluate Laplace transforms such as:
\(\large {\color{dodgerblue}{\mathscr{L} \left\{ t + e^{at} \right\} = \mbox{??}}}\)
\(\large {\color{dodgerblue}{\mathscr{L} \left\{ te^{at} \right\} =\mbox{??}}}\)
Properties of a Linear Operator
An operator (or transformation) \(T\) is a linear operator (or linear transformation) if for all \(f\) and \(g\) in the domain, \(T\) statisfies the following two properties:
- \(T( f + g) = T(f) + T(g)\).
- \(T( c f ) = cT(f)\), where \(c\) denotes a scalar.
Question 1:
Let \(f\) and \(g\) be functions whose Laplace transform exists for \(s > \alpha\) and let \(c\) be a constant. Answer parts a and b and show the Laplace transform is a linear operator.
Question 1a:
Prove the Laplace transform statisfies the additive property
\[{\large \color{dodgerblue}{\mathscr{L} \left\{ f + g \right\} = \mathscr{L} \left\{ f \right\} + \mathscr{L} \left\{ g \right\}}}.\]
Solution to Question 1a:
Question 1b:
Prove the Laplace transform satisfies the scalar multiplication property
\[{\color{dodgerblue}{\large \mathscr{L} \left\{ cf \right\} = c \mathscr{L} \left\{ f \right\}}}.\]
Solution to Question 1b:
Shifting Property: Laplace Transform of \(g(t)=e^{at}f(t)\)
Question 2:
If the Laplace transform \(\mathscr{L}\{ f(t) \} =F(s)\) exist for \(s > \alpha\), then show that
\[\color{dodgerblue}{\large \mathscr{L}\{ e^{at}f(t) \}= F(s-a), \ \ \mbox{for } s > \alpha + a}.\]
Solution to Question 2:
Question 3:
Using the shifting property from Question 2 and the fact that \(\mathscr{L} \{ \cos{(bt)} \} = \dfrac{s}{s^2+b^2}\) for \(s >0\), find \(\color{dodgerblue}{\large \mathscr{L} \left\{ e^{at} \cos{(bt)} \right\}}\).
Solution to Question 3:
Revisiting Laplace Transforms with SymPy
In Worksheet 20 Introduction to Laplace Transforms we used the SymPy library in Python to help verify and check our work with Laplace transforms.
- Run the first code cell below to create a Laplace transform function.
- Then run the second code cell to check your previous answer.
f = sym.exp(a*t) * sym.cos(b*t) # define your function with respect to t
L(f)Laplace Transform of Derivatives
A function is of exponential order \(\alpha\) if there exists positive constants \(C\) and \(T\) such that
\[\color{dodgerblue}{\large \left| f(t) \right| < Ce^{\alpha t} \ \ \mbox{for all } t > T}.\]
For example:
- \(f(t) = \cos{(5t)}e^{7t}\) has exponential order \(\alpha = 7\).
- \(f(t)=e^{t^2}\) does not have an exponential order.
Question 4:
If \(f(t)\) is continuous on \(\lbrack 0, \infty )\) and \(f'(t)\) is piecewise continuous on \(\lbrack 0, \infty )\) with both exponential order \(\alpha\), then prove for \(s > \alpha\),
\[\color{dodgerblue}{\large \mathscr{L} \left\{ f'(t) \right\} = s \mathscr{L} \left\{ f(t) \right\} - f(0) = sF(s)-f(0)}.\]
Solution to Question 4:
Question 5:
Using the property from Question 4 and the fact that \(\mathscr{L} \left\{ \cos{(bt)} \right\} = \frac{s}{s^2+b^2}\) for \(s >0\), find \(\color{dodgerblue}{\large \mathscr{L} \left\{ \sin{(bt)} \right\}}\).
Then check your answer using the Laplace transform function L(f) defined in Revisiting Laplace Transforms with SymPy.
Solution to Question 5:
After using the properties, verify your result using the code cell below.
#############################################################
# STUDENT TO DO:
# Replace the ?? with an approrpriate expression
# Be sure you have already created the L(f) fucntion
#############################################################
f = ?? # define your function with respect to t
L(f)Question 6:
If \(\mathscr{L} \left\{ f(t) \right\} = F(s)\) for all \(s > \alpha\), using the property from Question 4, show that
\[\color{dodgerblue}{\large \mathscr{L} \left\{ f''(t) \right\} =s^2F(s)-sf(0)-f'(0) \quad \mbox{for all } s > \alpha .}\]
Solution to Question 6:
Optional Proof Practice: Question 7
Using induction show that
\[\color{dodgerblue}{\large \mathscr{L} \left\{ f^{(n)} \right\} = s^n \mathscr{L} \left\{ f \right\} - s^{n-1} f(0) - s^{n-2} f'(0) - \ldots - f^{(n-1)}(0).}\]
Solution to Question 7:
Optional Proof Practice: Question 8
Let \(F(s) = \mathscr{L} \left\{ f \right\}\) and assume \(f(t)\) is piecewise continuous on \(\lbrack 0, \infty )\) and of exponential order \(\alpha\). Prove that for \(s > \alpha\) if follows that
\[\color{dodgerblue}{\large \mathscr{L} \left\{ t^nf(t) \right\} = (-1)^n \frac{d^nF}{ds^n}.}\]
Solution to Question 8:
Question 9:
Using the definition of the Laplace transform, the result that \(\mathscr{L} \left\{ e^{at} \right\} = \dfrac{1}{s-a}\) for \(s>a\) and the property above, find a formula for \(\color{dodgerblue}{\large \mathscr{L} \left\{ t^n e^{at} \right\}}\).
Then check your answer using the Laplace transform function L(f) defined in Revisiting Laplace Transforms with SymPy.
Solution to Question 9:
After using the properties, verify your result using the code cell below.
#############################################################
# STUDENT TO DO:
# Replace the ?? with an approrpriate expression
# Be sure you have already created the L(f) fucntion
#############################################################
n = sym.symbols('n', integer=True, positive=True)
f = ?? # define your function with respect to t
L(f)Common Laplace Transforms
Below is a table of updated table of common Laplace transforms pairs.
| \(\displaystyle \large f(t)\) | \(\displaystyle \large F(s) = \mathscr{L} \left\{ f(t) \right\}\) |
|---|---|
| \(\displaystyle \large f(t)=1\) | \(\displaystyle \large F(s)=\frac{1}{s}, \ s > 0\) |
| \(\displaystyle\large f(t)=e^{at}\) | \(\displaystyle \large F(s) = \frac{1}{s-a}, \ s > a\) |
| \(\displaystyle \large f(t)=t^n, \ n=1,2, \ldots\) | \(\displaystyle \large F(s) = \frac{n!}{s^{n+1}}, \ s > 0\) |
| \(\displaystyle \large f(t)=\sin{(bt)}\) | \(\displaystyle \large F(s) = \frac{b}{s^2+b^2}, \ s > 0\) |
| \(\displaystyle \large f(t)=\cos{(bt)}\) | \(\displaystyle \large F(s) = \frac{s}{s^2+b^2}, \ s > 0\) |
| \(\displaystyle \large e^{at}t^n, \ n=1,2, \ldots\) | \(\displaystyle \large F(s) = \frac{n!}{(s-a)^{n+1}}, \ s > a\) |
| \(\displaystyle \large e^{at}\sin{(bt)}\) | \(\displaystyle \large F(s) = \frac{b}{(s-a)^2+b^2}, \ s > a\) |
| \(\displaystyle \large e^{at}\cos{(bt)}\) | \(\displaystyle \large F(s) = \frac{s-a}{(s-a)^2+b^2}, \ s > a\) |
Properties of Laplace Transforms
1. \(\color{dodgerblue}{\mathscr{L} \left\{ cf(t) \right\} = c \mathscr{L} \left\{ f(t) \right\}}\), where \(c\) is a constant. See Question 1b.
2. \(\color{dodgerblue}{\mathscr{L} \left\{ f_1(t) + f_2(t) \right\} = \mathscr{L} \left\{ f_1(t) \right\} + \mathscr{L} \left\{ f_2(t)\right\}}\). See Question 1a.
3. If \(F(s) = \mathscr{L} \left\{ f(t) \right\}\) exists for all \(s > \alpha\), then \(\color{dodgerblue}{\displaystyle \mathscr{L} \left\{ e^{at} f(t) \right\} = F(s-a)}\) for all \(s>\alpha + a\). See Question 2.
4. If \(F(s) =\mathscr{L} \left\{ f(t) \right\}\) exists for all \(s > \alpha\), then for all \(s>\alpha\) (see Question 7),
\[\color{dodgerblue}{\mathscr{L} \left\{ f^{(n)}(t) \right\} = s^n \mathscr{L} \{ f(t) \}-s^{n-1} f(0)- s^{n-2} f'(0) - \ldots - f^{(n-1)}(0)}.\]
5. If \(F(s) =\mathscr{L} \left\{ f(t) \right\}\) exists for all \(s > \alpha\), then (see Question 8)
\[\color{dodgerblue}{\mathscr{L} \left\{ t^n f(t) \right\} = (-1)^n \frac{d^nF}{ds^n} \mbox{ for all } s > \alpha}.\]
Creative Commons License Information
Exploring Differential Equations by Adam Spiegler is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://github.com/CU-Denver-MathStats-OER/ODEs and original content created by Rasmussen, C., Keene, K. A., Dunmyre, J., & Fortune, N. (2018). Inquiry oriented differential equations: Course materials. Available at https://iode.sdsu.edu.