Using Laplace Transforms to Solve Initial Value Problems Bernd Schr - - PowerPoint PPT Presentation

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Using Laplace Transforms to Solve Initial Value Problems Bernd Schr - - PowerPoint PPT Presentation

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Overview An Example Double Check Using Laplace Transforms to Solve Initial Value Problems Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial

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SLIDE 1

logo1 Overview An Example Double Check

Using Laplace Transforms to Solve Initial Value Problems

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 2

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 3

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

Time Domain (t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 4

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

Time Domain (t)

Original DE & IVP

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 5

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

Time Domain (t)

Original DE & IVP ✲ L

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 6

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

Time Domain (t)

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 7

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

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 Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 8

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

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 Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 9

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

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 Using Laplace Transforms to Solve Initial Value Problems

slide-10
SLIDE 10

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

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 Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 11

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

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 Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 12

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 13

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 14

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 15

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 16

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-17
SLIDE 17

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0) 1.3 L

  • y′′

(s) = s2Y(s)−sy(0)−y′(0)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 18

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0) 1.3 L

  • y′′

(s) = s2Y(s)−sy(0)−y′(0) 1.4 L

  • y(n)(t)
  • (s) = snY(s)−sn−1y(0)−sn−2y′(0)−···−y(n−1)(0)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-19
SLIDE 19

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0) 1.3 L

  • y′′

(s) = s2Y(s)−sy(0)−y′(0) 1.4 L

  • y(n)(t)
  • (s) = snY(s)−sn−1y(0)−sn−2y′(0)−···−y(n−1)(0)
  • 2. The right sides above do not involve derivatives of

whatever Y is.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-20
SLIDE 20

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0) 1.3 L

  • y′′

(s) = s2Y(s)−sy(0)−y′(0) 1.4 L

  • y(n)(t)
  • (s) = snY(s)−sn−1y(0)−sn−2y′(0)−···−y(n−1)(0)
  • 2. The right sides above do not involve derivatives of

whatever Y is.

  • 3. The other key property is that constants and sums “factor

through” the Laplace transform:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-21
SLIDE 21

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0) 1.3 L

  • y′′

(s) = s2Y(s)−sy(0)−y′(0) 1.4 L

  • y(n)(t)
  • (s) = snY(s)−sn−1y(0)−sn−2y′(0)−···−y(n−1)(0)
  • 2. The right sides above do not involve derivatives of

whatever Y is.

  • 3. The other key property is that constants and sums “factor

through” the Laplace transform: L {f +g} = L {f}+L {g} and L {af} = aL {f}.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 22

logo1 Overview An Example Double Check

How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations

  • 1. The first key property of the Laplace transform is the way

derivatives are transformed.

1.1 L {y}(s) =: Y(s) (This is just notation.) 1.2 L

  • y′

(s) = sY(s)−y(0) 1.3 L

  • y′′

(s) = s2Y(s)−sy(0)−y′(0) 1.4 L

  • y(n)(t)
  • (s) = snY(s)−sn−1y(0)−sn−2y′(0)−···−y(n−1)(0)
  • 2. The right sides above do not involve derivatives of

whatever Y is.

  • 3. The other key property is that constants and sums “factor

through” the Laplace transform: L {f +g} = L {f}+L {g} and L {af} = aL {f}. (That is, the Laplace transform is linear.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 23

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 24

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 25

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 26

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 27

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2+7sY −7

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 28

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2+7sY −7+12Y =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

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SLIDE 29

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2+7sY −7+12Y =

  • s2 +7s+12
  • Y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-30
SLIDE 30

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2+7sY −7+12Y =

  • s2 +7s+12
  • Y

= s+9

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-31
SLIDE 31

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2+7sY −7+12Y =

  • s2 +7s+12
  • Y

= s+9 Y = s+9 s2 +7s+12

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-32
SLIDE 32

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Finding the Laplace transform of the solution. y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2 s2Y −s−2+7sY −7+12Y =

  • s2 +7s+12
  • Y

= s+9 Y = s+9 s2 +7s+12 = s+9 (s+3)(s+4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-33
SLIDE 33

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-34
SLIDE 34

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-35
SLIDE 35

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-36
SLIDE 36

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-37
SLIDE 37

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-38
SLIDE 38

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3) Heaviside′s Method :

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-39
SLIDE 39

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3) Heaviside′s Method : s = −3

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-40
SLIDE 40

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3) Heaviside′s Method : s = −3 A = 6

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-41
SLIDE 41

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3) Heaviside′s Method : s = −3 A = 6 s = −4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-42
SLIDE 42

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3) Heaviside′s Method : s = −3 A = 6 s = −4 B = −5

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-43
SLIDE 43

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Partial fraction decomposition. Y = s+9 (s+3)(s+4) = A s+3 + B s+4 s+9 (s+3)(s+4) = A(s+4)+B(s+3) (s+3)(s+4) s+9 = A(s+4)+B(s+3) Heaviside′s Method : s = −3 A = 6 s = −4 B = −5 Y = 6 s+3 − 5 s+4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-44
SLIDE 44

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Inverting the Laplace Transform.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-45
SLIDE 45

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Inverting the Laplace Transform. Y = 6 s+3 − 5 s+4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-46
SLIDE 46

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Inverting the Laplace Transform. Y = 6 s+3 − 5 s+4 Use the transform table.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-47
SLIDE 47

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Inverting the Laplace Transform. Y = 6 s+3 − 5 s+4 Use the transform table. L

  • eat

(s) = 1 s−a

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-48
SLIDE 48

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Inverting the Laplace Transform. Y = 6 s+3 − 5 s+4 Use the transform table. L

  • eat

(s) = 1 s−a = 6 1 s−(−3) −5 1 s−(−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-49
SLIDE 49

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

Inverting the Laplace Transform. Y = 6 s+3 − 5 s+4 Use the transform table. L

  • eat

(s) = 1 s−a = 6 1 s−(−3) −5 1 s−(−4) y = 6e−3t −5e−4t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-50
SLIDE 50

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2

y = 6e−3t −5e−4t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-51
SLIDE 51

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-52
SLIDE 52

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-53
SLIDE 53

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation. y′′ +7y′ +12y

?

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-54
SLIDE 54

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation. y′′ +7y′ +12y

?

=

  • 54e−3t−80e−4t

+7

  • −18e−3t+20e−4t

+12

  • 6e−3t−5e−4t

?

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-55
SLIDE 55

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation. y′′ +7y′ +12y

?

=

  • 54e−3t−80e−4t

+7

  • −18e−3t+20e−4t

+12

  • 6e−3t−5e−4t

?

= (54−126+72)e−3t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-56
SLIDE 56

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation. y′′ +7y′ +12y

?

=

  • 54e−3t−80e−4t

+7

  • −18e−3t+20e−4t

+12

  • 6e−3t−5e−4t

?

= (54−126+72)e−3t +(−80+140−60)e−4t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-57
SLIDE 57

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation. y′′ +7y′ +12y

?

=

  • 54e−3t−80e−4t

+7

  • −18e−3t+20e−4t

+12

  • 6e−3t−5e−4t

?

= (54−126+72)e−3t +(−80+140−60)e−4t

?

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-58
SLIDE 58

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the differential equation. y′′ +7y′ +12y

?

=

  • 54e−3t−80e−4t

+7

  • −18e−3t+20e−4t

+12

  • 6e−3t−5e−4t

?

= (54−126+72)e−3t +(−80+140−60)e−4t

?

=

√

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-59
SLIDE 59

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-60
SLIDE 60

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-61
SLIDE 61

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-62
SLIDE 62

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5 = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-63
SLIDE 63

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5 = 1 √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-64
SLIDE 64

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5 = 1 √ y′ = −18e−3t +20e−4t

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-65
SLIDE 65

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5 = 1 √ y′ = −18e−3t +20e−4t y′(0) = −18+20

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-66
SLIDE 66

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5 = 1 √ y′ = −18e−3t +20e−4t y′(0) = −18+20 = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

slide-67
SLIDE 67

logo1 Overview An Example Double Check

Does y = 6e−3t −5e−4t Really Solve the Initial Value Problem y′′ +7y′ +12y = 0, y(0) = 1, y′(0) = 2?

Checking the initial values. y = 6e−3t −5e−4t y(0) = 6−5 = 1 √ y′ = −18e−3t +20e−4t y′(0) = −18+20 = 2 √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems