Laplace Transforms and Convolutions Bernd Schr oder logo1 Bernd - - PowerPoint PPT Presentation

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Laplace Transforms and Convolutions Bernd Schr oder logo1 Bernd - - PowerPoint PPT Presentation

Feb 01, 2023 295 likes •818 views

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Laplace Transforms and Convolutions Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Laplace

slide-1
SLIDE 1

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Laplace Transforms and Convolutions

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-2
SLIDE 2

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-3
SLIDE 3

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-4
SLIDE 4

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-5
SLIDE 5

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t)

Original DE & IVP

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-6
SLIDE 6

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t)

Original DE & IVP ✲ L

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-7
SLIDE 7

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t)

Original DE & IVP Algebraic equation for the Laplace transform ✲ L

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-8
SLIDE 8

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t) Transform domain (s)

Original DE & IVP Algebraic equation for the Laplace transform ✲ L

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-9
SLIDE 9

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t) Transform domain (s)

Original DE & IVP Algebraic equation for the Laplace transform ✲ L Algebraic solution, partial fractions ❄

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-10
SLIDE 10

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t) Transform domain (s)

Original DE & IVP Algebraic equation for the Laplace transform Laplace transform

  • f the solution

✲ L Algebraic solution, partial fractions ❄

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-11
SLIDE 11

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t) Transform domain (s)

Original DE & IVP Algebraic equation for the Laplace transform Laplace transform

  • f the solution

✲ ✛ L L −1 Algebraic solution, partial fractions ❄

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-12
SLIDE 12

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Everything Remains As It Was

No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t) Transform domain (s)

Original DE & IVP Algebraic equation for the Laplace transform Laplace transform

  • f the solution

Solution ✲ ✛ L L −1 Algebraic solution, partial fractions ❄

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-13
SLIDE 13

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-14
SLIDE 14

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

  • 1. Solving initial value problems ay′′ +by′ +cy = f with

Laplace transforms leads to a transform Y = F ·R(s)+···.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-15
SLIDE 15

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

  • 1. Solving initial value problems ay′′ +by′ +cy = f with

Laplace transforms leads to a transform Y = F ·R(s)+···.

  • 2. If the Laplace transform F of f is not easily computed or if

the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-16
SLIDE 16

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

  • 1. Solving initial value problems ay′′ +by′ +cy = f with

Laplace transforms leads to a transform Y = F ·R(s)+···.

  • 2. If the Laplace transform F of f is not easily computed or if

the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a

  • product. Maybe that way the transformation of f can be

avoided.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-17
SLIDE 17

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

  • 1. Solving initial value problems ay′′ +by′ +cy = f with

Laplace transforms leads to a transform Y = F ·R(s)+···.

  • 2. If the Laplace transform F of f is not easily computed or if

the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a

  • product. Maybe that way the transformation of f can be

avoided.

  • 3. The convolution of the functions f(t) and g(t) is

f ∗g(t) =

t

0 f(τ)g(t −τ) dτ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-18
SLIDE 18

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

  • 1. Solving initial value problems ay′′ +by′ +cy = f with

Laplace transforms leads to a transform Y = F ·R(s)+···.

  • 2. If the Laplace transform F of f is not easily computed or if

the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a

  • product. Maybe that way the transformation of f can be

avoided.

  • 3. The convolution of the functions f(t) and g(t) is

f ∗g(t) =

t

0 f(τ)g(t −τ) dτ and L (f ∗g) = FG.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-19
SLIDE 19

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Inverse Laplace Transform of a Product

  • 1. Solving initial value problems ay′′ +by′ +cy = f with

Laplace transforms leads to a transform Y = F ·R(s)+···.

  • 2. If the Laplace transform F of f is not easily computed or if

the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a

  • product. Maybe that way the transformation of f can be

avoided.

  • 3. The convolution of the functions f(t) and g(t) is

f ∗g(t) =

t

0 f(τ)g(t −τ) dτ and L (f ∗g) = FG.

  • 4. So it is possible to avoid transforming the forcing term, but

the price we pay is that the solution is represented as an integral.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-20
SLIDE 20

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-21
SLIDE 21

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-22
SLIDE 22

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

natural response

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-23
SLIDE 23

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

natural response r(t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-24
SLIDE 24

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

natural response r(t) forcing function

✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-25
SLIDE 25

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

natural response r(t) forcing function p(t)

✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-26
SLIDE 26

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

natural response r(t) forcing function p(t)

  • verall

response

✲ ✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-27
SLIDE 27

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

The Convolution Can Be Useful When Larger Systems are Analyzed

natural response r(t) forcing function p(t) p(t)∗r(t)

  • verall

response

✲ ✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-28
SLIDE 28

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-29
SLIDE 29

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-30
SLIDE 30

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0 3y′ +2y = p(t), y(0) = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-31
SLIDE 31

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0 3y′ +2y = p(t), y(0) = 0 3sY +2Y = P

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-32
SLIDE 32

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0 3y′ +2y = p(t), y(0) = 0 3sY +2Y = P Y = P 1 3s+2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-33
SLIDE 33

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0 3y′ +2y = p(t), y(0) = 0 3sY +2Y = P Y = P 1 3s+2 = P1 3 1 s+ 2

3

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-34
SLIDE 34

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0 3y′ +2y = p(t), y(0) = 0 3sY +2Y = P Y = P 1 3s+2 = P1 3 1 s+ 2

3

y(t) = p(t)∗ 1 3e− 2

3t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-35
SLIDE 35

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Solve the Initial Value Problem 3y′ +2y =

  • sin(t)
  • , y(0) = 0

3y′ +2y =

  • sin(t)
  • ,

y(0) = 0 3y′ +2y = p(t), y(0) = 0 3sY +2Y = P Y = P 1 3s+2 = P1 3 1 s+ 2

3

y(t) = p(t)∗ 1 3e− 2

3t = 1

3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-36
SLIDE 36

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-37
SLIDE 37

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-38
SLIDE 38

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y!

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-39
SLIDE 39

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-40
SLIDE 40

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-41
SLIDE 41

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-42
SLIDE 42

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-43
SLIDE 43

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-44
SLIDE 44

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • +
  • −2

3 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-45
SLIDE 45

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • +
  • −2

3 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • − 2

3y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-46
SLIDE 46

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • +
  • −2

3 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • − 2

3y 3y′ +2y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-47
SLIDE 47

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • +
  • −2

3 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • − 2

3y 3y′ +2y = 3 1 3

  • sin(t)
  • − 2

3y

  • +2y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-48
SLIDE 48

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • +
  • −2

3 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • − 2

3y 3y′ +2y = 3 1 3

  • sin(t)
  • − 2

3y

  • +2y =
  • sin(t)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-49
SLIDE 49

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Does y = 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ Really Solve the Initial Value Problem

3y′ +2y =

  • sin(t)
  • , y(0) = 0 ?

Initial value: Look at y! y′ = d dt 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • e− 2

3(t−t) + 1

3

t

  • sin(τ)
  • ∂

∂te− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • +
  • −2

3 1 3

t

  • sin(τ)
  • e− 2

3(t−τ) dτ

= 1 3

  • sin(t)
  • − 2

3y 3y′ +2y = 3 1 3

  • sin(t)
  • − 2

3y

  • +2y =
  • sin(t)
  • √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-50
SLIDE 50

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Comparing Output to Input

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-51
SLIDE 51

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Comparing Output to Input

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

slide-52
SLIDE 52

logo1 Transforms and New Formulas Using Convolutions An Example Double Check Visualization

Comparing Output to Input

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions