[PPT] - Laplace Transforms of Periodic Functions Bernd Schr oder logo1 PowerPoint Presentation

SLIDE 1

logo1 Transforms and New Formulas An Example Double Check Visualization

Laplace Transforms of Periodic Functions

Bernd Schr¨

der

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 2

logo1 Transforms and New Formulas An Example Double Check Visualization

Everything Remains As It Was

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 3

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 4

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 5

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 6

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 7

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 8

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 9

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 10

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 11

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 12

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 13

logo1 Transforms and New Formulas An Example Double Check Visualization

Periodic Functions

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 14

logo1 Transforms and New Formulas An Example Double Check Visualization

Periodic Functions

1. A function f is periodic with period T > 0 if and only if for

all t we have f(t +T) = f(t).

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 15

logo1 Transforms and New Formulas An Example Double Check Visualization

Periodic Functions

1. A function f is periodic with period T > 0 if and only if for

all t we have f(t +T) = f(t).

2. If f is bounded, piecewise continuous and periodic with

period T, then L

f(t)
=

1 1−e−sT

T

0 e−stf(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 16

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 17

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 18

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 19

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt = ∞

∑

n=0

(n+1)T

nT

e−stf(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 20

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt = ∞

∑

n=0

(n+1)T

nT

e−stf(t) dt =

∞

∑

n=0

(n+1)T

nT

e−s

(t−nT)+nT
f(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 21

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt = ∞

∑

n=0

(n+1)T

nT

e−stf(t) dt =

∞

∑

n=0

(n+1)T

nT

e−s

(t−nT)+nT
f(t) dt =

∞

∑

n=0

T

0 e−s(u+nT)f(u) du

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 22

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt = ∞

∑

n=0

(n+1)T

nT

e−stf(t) dt =

∞

∑

n=0

(n+1)T

nT

e−s

(t−nT)+nT
f(t) dt =

∞

∑

n=0

T

0 e−s(u+nT)f(u) du

=

∞

∑

n=0

e−nsT

T

0 e−suf(u) du

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 23

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt = ∞

∑

n=0

(n+1)T

nT

e−stf(t) dt =

∞

∑

n=0

(n+1)T

nT

e−s

(t−nT)+nT
f(t) dt =

∞

∑

n=0

T

0 e−s(u+nT)f(u) du

=

∞

∑

n=0

e−nsT

T

0 e−suf(u) du =

∞

∑

n=0

e−sTn

T

0 e−stf(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 24

logo1 Transforms and New Formulas An Example Double Check Visualization

How Did We Get That?

L

f(t)
=

∞

0 e−stf(t) dt = ∞

∑

n=0

(n+1)T

nT

e−stf(t) dt =

∞

∑

n=0

(n+1)T

nT

e−s

(t−nT)+nT
f(t) dt =

∞

∑

n=0

T

0 e−s(u+nT)f(u) du

=

∞

∑

n=0

e−nsT

T

0 e−suf(u) du =

∞

∑

n=0

e−sTn

T

0 e−stf(t) dt

= 1 1−e−sT

T

0 e−stf(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 25

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 26

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 27

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 28

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

= 1 1−e−sπ

∞

0 e−st

1−U (t −π)

sin(t) dt

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 29

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

= 1 1−e−sπ

∞

0 e−st

1−U (t −π)

sin(t) dt

= 1 1−e−sπ

L
sin(t)
Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 30

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

= 1 1−e−sπ

∞

0 e−st

1−U (t −π)

sin(t) dt

= 1 1−e−sπ

L
sin(t)
+L
U (t −π)sin(t −π)
Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 31

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

= 1 1−e−sπ

∞

0 e−st

1−U (t −π)

sin(t) dt

= 1 1−e−sπ

L
sin(t)
+L
U (t −π)sin(t −π)
=

1 1−e−sπ

1

s2 +1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 32

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

= 1 1−e−sπ

∞

0 e−st

1−U (t −π)

sin(t) dt

= 1 1−e−sπ

L
sin(t)
+L
U (t −π)sin(t −π)
=

1 1−e−sπ

1

s2 +1 +e−πs 1 s2 +1

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 33

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

L

|sin(t)|
=

1 1−e−sπ

π

0 e−st

sin(t)

dt

= 1 1−e−sπ

π

0 e−st sin(t) dt

= 1 1−e−sπ

∞

0 e−st

1−U (t −π)

sin(t) dt

= 1 1−e−sπ

L
sin(t)
+L
U (t −π)sin(t −π)
=

1 1−e−sπ

1

s2 +1 +e−πs 1 s2 +1

=

1 1−e−sπ

1+e−πs

1 s2 +1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 34

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 35

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

3sY +2Y

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 36

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

3sY +2Y = 1 1−e−sπ

1+e−πs

1 s2 +1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 37

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

3sY +2Y = 1 1−e−sπ

1+e−πs

1 s2 +1 Y = 1 1−e−sπ

1+e−πs

1 (s2 +1)(3s+2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 38

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

3sY +2Y = 1 1−e−sπ

1+e−πs

1 s2 +1 Y = 1 1−e−sπ

1+e−πs

1 (s2 +1)(3s+2) =

∞

∑

n=0

e−πsn

1+e−πs 1 (s2 +1)(3s+2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 39

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

3sY +2Y = 1 1−e−sπ

1+e−πs

1 s2 +1 Y = 1 1−e−sπ

1+e−πs

1 (s2 +1)(3s+2) =

∞

∑

n=0

e−πsn

1+e−πs 1 (s2 +1)(3s+2) =

∞

∑

n=0

e−πsn +

∞

∑

n=0

e−πsn+1
1

(s2 +1)(3s+2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 40

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

3sY +2Y = 1 1−e−sπ

1+e−πs

1 s2 +1 Y = 1 1−e−sπ

1+e−πs

1 (s2 +1)(3s+2) =

∞

∑

n=0

e−πsn

1+e−πs 1 (s2 +1)(3s+2) =

∞

∑

n=0

e−πsn +

∞

∑

n=0

e−πsn+1
1

(s2 +1)(3s+2) =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

(s2 +1)(3s+2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 41

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 42

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 43

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 44

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 :

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 45

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 46

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 47

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 48

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 :

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 49

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 50

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 51

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 52

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 :

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 53

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 54

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 55

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13, A = − 3 13

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 56

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13, A = − 3 13 1 (s2 +1)(3s+2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 57

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13, A = − 3 13 1 (s2 +1)(3s+2) = 1 13

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 58

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13, A = − 3 13 1 (s2 +1)(3s+2) = 1 13 −3s+2 s2 +1

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 59

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13, A = − 3 13 1 (s2 +1)(3s+2) = 1 13 −3s+2 s2 +1 + 9 3s+2

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 60

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

1 (s2 +1)(3s+2) = As+B s2 +1 + C 3s+2 1 = (As+B)(3s+2)+C

s2 +1
s = −2

3 : 1 = C 4 9 +1

= 13

9 C, C = 9 13 s = 0 : 1 = B·2+C ·1 = 2B+ 9 13, B = 2 13 s = 1 : 1 = (A+B)·5+C ·2 = 5A+ 10 13 + 18 13, A = − 3 13 1 (s2 +1)(3s+2) = 1 13 −3s+2 s2 +1 + 9 3s+2

=

1 13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 61

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 62

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

Y =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

(s2 +1)(3s+2)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 63

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

Y =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

(s2 +1)(3s+2) =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 64

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

Y =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

(s2 +1)(3s+2) =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

=
∞

∑

n=0

e−nπs +

∞

∑

n=1

e−nπs

1

13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 65

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

Y =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

(s2 +1)(3s+2) =

∞

∑

n=0

e−nπs +

∞

∑

n=0

e−(n+1)πs

1

13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

=
∞

∑

n=0

e−nπs +

∞

∑

n=1

e−nπs

1

13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

=
1+2

∞

∑

n=1

e−nπs

1

13

−3

s s2 +1 +2 1 s2 +1 +3 1 s+ 2

3

Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 66

logo1 Transforms and New Formulas 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 of Periodic Functions

SLIDE 67

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

y = 1 13

−3cos(t)+2sin(t)+3e− 2

3 t

+ 2 13

∞

∑

n=1

U

t −nπ
−3cos(t)+2sin(t)+3e− 2

3 t

t→t−nπ

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 68

logo1 Transforms and New Formulas An Example Double Check Visualization

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

sin(t)
, y(0) = 0

y = 1 13

−3cos(t)+2sin(t)+3e− 2

3 t

+ 2 13

∞

∑

n=1

U

t −nπ
−3cos(t)+2sin(t)+3e− 2

3 t

t→t−nπ

= 1 13

−3cos(t)+2sin(t)+3e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

−3cos(t −nπ)+2sin(t −nπ)+3e− 2

3 (t−nπ) Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 69

logo1 Transforms and New Formulas An Example Double Check Visualization

Does y = 1 13

−3cos(t)+2sin(t)+3e− 2

3t

+ 2 13

∞

∑

n=1

U (t −nπ)

−3cos(t −nπ)+2sin(t −nπ)+3e− 2

3 (t−nπ)

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 of Periodic Functions

SLIDE 70

logo1 Transforms and New Formulas An Example Double Check Visualization

Does y = 1 13

−3cos(t)+2sin(t)+3e− 2

3t

+ 2 13

∞

∑

n=1

U (t −nπ)

−3cos(t −nπ)+2sin(t −nπ)+3e− 2

3 (t−nπ)

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

sin(t)
, y(0) = 0?

y(0)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 71

logo1 Transforms and New Formulas An Example Double Check Visualization

Does y = 1 13

−3cos(t)+2sin(t)+3e− 2

3t

+ 2 13

∞

∑

n=1

U (t −nπ)

−3cos(t −nπ)+2sin(t −nπ)+3e− 2

3 (t−nπ)

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

sin(t)
, y(0) = 0?

y(0) = 1 13

−3cos(0)+2sin(0)+3e− 2

30

+ 2 13

∞

∑

n=1

U (0−nπ)

−3cos(0−nπ)+2sin(0−nπ)+3e− 2

3 (0−nπ) Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 72

logo1 Transforms and New Formulas An Example Double Check Visualization

Does y = 1 13

−3cos(t)+2sin(t)+3e− 2

3t

+ 2 13

∞

∑

n=1

U (t −nπ)

−3cos(t −nπ)+2sin(t −nπ)+3e− 2

3 (t−nπ)

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

sin(t)
, y(0) = 0?

y(0) = 1 13

−3cos(0)+2sin(0)+3e− 2

30

+ 2 13

∞

∑

n=1

U (0−nπ)

−3cos(0−nπ)+2sin(0−nπ)+3e− 2

3 (0−nπ)

=

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 73

logo1 Transforms and New Formulas An Example Double Check Visualization

Does y = 1 13

−3cos(t)+2sin(t)+3e− 2

3t

+ 2 13

∞

∑

n=1

U (t −nπ)

−3cos(t −nπ)+2sin(t −nπ)+3e− 2

3 (t−nπ)

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

sin(t)
, y(0) = 0?

y(0) = 1 13

−3cos(0)+2sin(0)+3e− 2

30

+ 2 13

∞

∑

n=1

U (0−nπ)

−3cos(0−nπ)+2sin(0−nπ)+3e− 2

3 (0−nπ)

= √

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 74

logo1 Transforms and New Formulas An Example Double Check Visualization Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 75

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ) Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 76

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ)

3y′ +2y = 1 13

9sin(t)+6cos(t)−6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

9sin(t −nπ)+6cos(t −nπ)−6e− 2

3 (t−nπ) Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 77

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ)

3y′ +2y = 1 13

9sin(t)+6cos(t)−6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

9sin(t −nπ)+6cos(t −nπ)−6e− 2

3 (t−nπ)

+ 1 13

−6cos(t)+4sin(t)+6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

−6cos(t −nπ)+4sin(t −nπ)+6e− 2

3 (t−nπ) Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 78

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ)

3y′ +2y = 1 13

9sin(t)+6cos(t)−6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

9sin(t −nπ)+6cos(t −nπ)−6e− 2

3 (t−nπ)

+ 1 13

−6cos(t)+4sin(t)+6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

−6cos(t −nπ)+4sin(t −nπ)+6e− 2

3 (t−nπ)

= 1 13 [9sin(t)+4sin(t)]+ 2 13

∞

∑

n=1

U (t −nπ)[9sin(t −nπ)+4sin(t −nπ)]

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 79

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ)

3y′ +2y = 1 13

9sin(t)+6cos(t)−6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

9sin(t −nπ)+6cos(t −nπ)−6e− 2

3 (t−nπ)

+ 1 13

−6cos(t)+4sin(t)+6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

−6cos(t −nπ)+4sin(t −nπ)+6e− 2

3 (t−nπ)

= 1 13 [9sin(t)+4sin(t)]+ 2 13

∞

∑

n=1

U (t −nπ)[9sin(t −nπ)+4sin(t −nπ)] = sin(t)+2

∞

∑

n=1

U (t −nπ)sin(t −nπ)

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 80

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ)

3y′ +2y = 1 13

9sin(t)+6cos(t)−6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

9sin(t −nπ)+6cos(t −nπ)−6e− 2

3 (t−nπ)

+ 1 13

−6cos(t)+4sin(t)+6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

−6cos(t −nπ)+4sin(t −nπ)+6e− 2

3 (t−nπ)

= 1 13 [9sin(t)+4sin(t)]+ 2 13

∞

∑

n=1

U (t −nπ)[9sin(t −nπ)+4sin(t −nπ)] = sin(t)+2

∞

∑

n=1

U (t −nπ)sin(t −nπ) =

sin(t)
Bernd Schr¨
der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 81

logo1 Transforms and New Formulas An Example Double Check Visualization

y′ = 1 13

3sin(t)+2cos(t)−2e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

3sin(t −nπ)+2cos(t −nπ)−2e− 2

3 (t−nπ)

3y′ +2y = 1 13

9sin(t)+6cos(t)−6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

9sin(t −nπ)+6cos(t −nπ)−6e− 2

3 (t−nπ)

+ 1 13

−6cos(t)+4sin(t)+6e− 2

3 t

+ 2 13

∞

∑

n=1

U (t −nπ)

−6cos(t −nπ)+4sin(t −nπ)+6e− 2

3 (t−nπ)

= 1 13 [9sin(t)+4sin(t)]+ 2 13

∞

∑

n=1

U (t −nπ)[9sin(t −nπ)+4sin(t −nπ)] = sin(t)+2

∞

∑

n=1

U (t −nπ)sin(t −nπ) =

sin(t)
√

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 82

logo1 Transforms and New Formulas An Example Double Check Visualization

Comparing Output to Input

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 83

logo1 Transforms and New Formulas An Example Double Check Visualization

Comparing Output to Input

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions

SLIDE 84

logo1 Transforms and New Formulas An Example Double Check Visualization

Comparing Output to Input

Bernd Schr¨

der

Louisiana Tech University, College of Engineering and Science Laplace Transforms of Periodic Functions