Laplace Transform Periodic
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Transforms and New Formulas An Example Double Check Visualization
Laplace Transforms of Periodic Functions
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
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Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
-
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
-
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Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
-
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Original
DE & IVP
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
-
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Original
DE & IVP
L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
T f d N F l A E l D bl Ch k Vi li ti
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Original
DE & IVP
Algebraic equation for
the Laplace transform
L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
T f d N F l A E l D bl Ch k Vi li ti
-
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8/84
logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
Original
DE & IVP
Algebraic equation for
the Laplace transformL
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
Original
DE & IVP
Algebraic equation for
the Laplace transformL
Algebraic solution,
partial fractions
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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10/84
logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
Original
DE & IVP
Algebraic equation for
the Laplace transform
Laplace transform
of the solution
L
Algebraic solution,
partial fractions
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
Original
DE & IVP
Algebraic equation for
the Laplace transform
Laplace transform
of the solution
L
L1
Algebraic solution,
partial fractions
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Transforms and New Formulas An Example Double Check Visualization
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
Original
DE & IVP
Algebraic equation for
the Laplace transform
Laplace transform
of the solutionSolution
L
L1
Algebraic solution,
partial fractions
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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p
Periodic Functions
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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p
Periodic Functions1. A functionfis periodic with periodT>0 if and only if for
alltwe havef(t+T) =f(t).
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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Periodic Functions1. A functionfis periodic with periodT>0 if and only if for
alltwe havef(t+T) =f(t).2. Iffis bounded, piecewise continuous and periodic with
periodT, then
Lf(t)
=
1
1 esT T
0 est
f(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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How Did We Get That?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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How Did We Get That?
Lf(t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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How Did We Get That?
Lf(t)
=
0
estf(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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How Did We Get That?
Lf(t)
=
0
estf(t)dt=
n=0
(n+1)TnT
estf(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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How Did We Get That?
Lf(t)
=
0
estf(t)dt=
n=0
(n+1)TnT
estf(t)dt
=
n=0
(n+1)T
nT
es(tnT)+nTf(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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How Did We Get That?
Lf(t)
=
0
estf(t)dt=
n=0
(n+1)TnT
estf(t)dt
=
n=0
(n+1)T
nT
es(tnT)+nTf(t)dt=
n=0
T
0
es(u+nT)f(u)du
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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How Did We Get That?
Lf(t)
=
0estf(t)dt=
n=0
(n+1)TnT
estf(t)dt
=
n=0
(n+1)T
nT
es(tnT)+nTf(t)dt=
n=0
T
0
es(u+nT)f(u)du
=
n=0
ensT T
0esuf(u)du
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
How Did We Get That?
Lf(t)
=
0estf(t)dt=
n=0
(n+1)TnT
estf(t)dt
=
n=0
(n+1)T
nT
es(tnT)+nTf(t)dt=
n=0
T
0
es(u+nT)f(u)du
=
n=0
ensT T
0esuf(u)du=
n=0
esT
n T0
estf(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
How Did We Get That?
Lf(t)
=
0estf(t)dt=
n=0
(n+1)TnT
estf(t)dt
=
n=0
(n+1)T
nT
es(tnT)+nTf(t)dt=
n=0
T
0
es(u+nT)f(u)du
=
n=0
ensT T
0esuf(u)du=
n=0
esT
n T0
estf(t)dt
=
1
1 esT T
0 est
f(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0
estsin(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
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Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0
estsin(t)dt
= 1
1 es
0est
1U(t)sin(t)dt
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0
estsin(t)dt
= 1
1 es
0est
1U(t)sin(t)dt
= 1
1
es L sin(t)
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0estsin(t)dt
= 1
1 es
0est
1U(t)sin(t)dt
= 1
1
es L sin(t)+LU(t) sin(t)
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0estsin(t)dt
= 1
1 es
0est
1U(t)sin(t)dt
= 1
1
es L sin(t)+LU(t) sin(t)
= 1
1 es
1
s2 + 1
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0estsin(t)dt
= 1
1 es
0est
1U(t)sin(t)dt
= 1
1
es L sin(t)+LU(t) sin(t)
= 1
1 es
1
s2 + 1+ es 1
s2 + 1
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
L|sin(t)| = 1
1 es
0est
sin(t)dt=
1
1
es
0estsin(t)dt
= 1
1 es
0est
1U(t)sin(t)dt
= 1
1
es L sin(t)+LU(t) sin(t)
= 1
1 es
1
s2 + 1+ es 1
s2 + 1
= 1
1
es
1 + es
1
s2 + 1Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
Bernd Schroder Louisiana Tech University, College of Engineering and ScienceLaplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
3sY+ 2Y
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
3sY+ 2Y = 1
1 es
1 + es 1s2 + 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
3sY+ 2Y = 1
1 es
1 + es 1s2 + 1
Y = 1
1
es 1 + e
s
1
(s2 + 1) (3s+ 2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
3sY+ 2Y = 1
1 es
1 + es 1s2 + 1
Y = 1
1
es 1 + e
s
1
(s2 + 1) (3s+ 2)
=
n=0
es
n 1 + es
1(s2 + 1) (3s+ 2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
3sY+ 2Y = 1
1 es
1 + es 1s2 + 1
Y = 1
1
es 1 + e
s
1
(s2 + 1) (3s+ 2)
=
n=0
es
n 1 + es
1(s2 + 1) (3s+ 2)
=
n=0
esn +
n=0
esn+1 1(s2 + 1) (3s+ 2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem
3y+ 2y=
sin(t),y(0) =0
3sY+ 2Y = 1
1 es
1 + es 1s2 + 1
Y = 1
1
es 1 + e
s
1
(s2 + 1) (3s+ 2)
=
n=0
es
n 1 + es
1(s2 + 1) (3s+ 2)
=
n=0
esn +
n=0
esn+1 1(s2 + 1) (3s+ 2)
=
n=0
ens +
n=0
e(n+1)s
1
(s2 + 1) (3s+ 2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
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 Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3:
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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45/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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46/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
47/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
48/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 :
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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49/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
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50/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2) =
As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
51/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2)
= As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
52/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2)
= As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 :
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
53/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2 + 1) (3s+ 2)
= As+B
s2 + 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
54/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1 +
C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
( ) ( )
-
8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1
+ C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13, A= 3
13
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
S l h I i i l V l P bl 3 2
i ( )
(0) 0
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1
+ C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13, A= 3
13
1
(s2 + 1) (3s+ 2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
S l h I i i l V l P bl 3 2
i ( )
(0) 0
-
8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1
+ C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13, A= 3
13
1
(s2 + 1) (3s+ 2) =
1
13
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
S l th I iti l V l P bl 3 + 2
i (t)
(0) 0
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8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1
+ C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13, A= 3
13
1
(s2 + 1) (3s+ 2) =
1
133s+ 2
s2 + 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Sol e the Initial Val e Problem 3 + 2
in(t)
(0) 0
-
8/11/2019 Laplace Transform Periodic
59/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1
+ C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13, A= 3
13
1
(s2 + 1) (3s+ 2) =
1
133s+ 2
s2 + 1 +
9
3s+ 2
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y + 2y
sin(t)y(0) 0
-
8/11/2019 Laplace Transform Periodic
60/84
logo1
Solve the Initial Value Problem 3y+ 2y=
sin(t),y(0) =0
1
(s2
+ 1) (3s+ 2)
= As+B
s2
+ 1
+ C
3s+ 21 = (As+B)(3s+ 2) +C
s2 + 1
s=2
3: 1 = C
4
9+ 1
=
13
9C, C=
9
13
s=0 : 1 = B 2 +C1=2B+ 913
, B= 213
s=1 : 1 = (A+B) 5 +C2=5A+1013
+18
13, A= 3
13
1
(s2 + 1) (3s+ 2) =
1
133s+ 2
s2 + 1 +
9
3s+ 2
= 1
13
3 s
s2 + 1+ 2
1
s2 + 1+ 3
1
s+ 23
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem
-
8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem
3y+ 2y= sin(t),y(0) =0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem
-
8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem
3y+ 2y= sin(t),y(0) =0Y =
n=0
ens +
n=0
e(n+1)s
1
(s2 + 1) (3s+ 2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem
-
8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem
3y+ 2y= sin(t),y(0) =0Y =
n=0
ens +
n=0
e(n+1)s
1
(s2 + 1) (3s+ 2)
= n=0e
ns
+
n=0e(n+1)s 1
133
s
s2 + 1 + 2
1
s2 + 1 + 3
1
s+ 23
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem
-
8/11/2019 Laplace Transform Periodic
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logo1
Solve the Initial Value Problem
3y+ 2y= sin(t),y(0) =0Y =
n=0
ens +
n=0
e(n+1)s
1
(s2 + 1) (3s+ 2)
= n=0e
ns
+
n=0e(n+1)s 1
133
s
s2 + 1 + 2
1
s2 + 1 + 3
1
s+ 23
=
n=0
ens +
n=1
ens
1
13
3 s
s2 + 1+ 2
1
s2 + 1+ 3
1
s+ 23
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem
-
8/11/2019 Laplace Transform Periodic
65/84
logo1
Solve the Initial Value Problem
3y+ 2y= sin(t),y(0) =0Y =
n=0
ens +
n=0
e(n+1)s
1
(s2 + 1) (3s+ 2)
= n=0e
ns
+
n=0e(n+1)s 1
133
s
s2 + 1 + 2
1
s2 + 1 + 3
1
s+ 23
=
n=0
ens +
n=1
ens
1
13
3 s
s2 + 1+ 2
1
s2 + 1+ 3
1
s+ 23
=
1 + 2n=1
ens
1
13
3 s
s2 + 1+ 2 1
s2 + 1+ 3 1
s+ 23
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y + 2y =sin(t), y(0) = 0
-
8/11/2019 Laplace Transform Periodic
66/84
logo1
Solve the Initial Value Problem 3y + 2y sin(t),y(0) 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y+ 2y=sin(t),y(0) =0
-
8/11/2019 Laplace Transform Periodic
67/84
logo1
y + y ( ), y( )y =
1
13 3cos(t) + 2sin(t) + 3e 2
3t
+
2
13
n=1
Utn3cos(t) + 2sin(t) + 3e 23 t
ttn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y+ 2y=sin(t),y(0) =0
-
8/11/2019 Laplace Transform Periodic
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logo1
y y ( ), y( )y =
1
13 3cos(t) + 2sin(t) + 3e 2
3t
+
2
13
n=1
Utn3cos(t) + 2sin(t) + 3e 23 t
ttn
= 1
13
3cos(t) + 2sin(t) + 3e 23 t
+ 213
n=1
U(tn)3cos(tn) + 2sin(tn) + 3e 23 (tn)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Doesy= 1
13
3cos(t) + 2sin(t) + 3e 23 t
-
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logo1
y13
( ) ( )
+
2
13
n=1
U(t
n)3cos(tn) + 2sin(tn) + 3e 2
3(tn)Really Solve
the Initial Value Problem 3y+ 2y=sin(t),y(0) =0?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Doesy= 1
13
3cos(t) + 2sin(t) + 3e 23 t
-
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logo1
y13
( ) ( )
+
2
13
n=1
U(t
n)3cos(tn) + 2sin(tn) + 3e 2
3(tn)Really Solve
the Initial Value Problem 3y+ 2y=sin(t),y(0) =0?
y(0)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
-
8/11/2019 Laplace Transform Periodic
71/84
Transforms and New Formulas An Example Double Check Visualization
Doesy= 1
13
3cos(t) + 2sin(t) + 3e 23 t
-
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logo1
13
+
2
13
n=1
U(t
n)3cos(tn) + 2sin(tn) + 3e
23
(tn)Really Solvethe Initial Value Problem 3y+ 2y=
sin(t),y(0) =0?y(0) =
1
133cos(0) + 2sin(0) + 3e
23
0+
2
13
n=1
U(0n)3cos(0n) + 2sin(0n) + 3e 23 (0n)
= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Doesy= 1
13
3cos(t) + 2sin(t) + 3e 23 t
-
8/11/2019 Laplace Transform Periodic
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logo1
13
+
2
13
n=1
U(t
n)3cos(tn) + 2sin(tn) + 3e
23
(tn)Really Solvethe Initial Value Problem 3y+ 2y=
sin(t),y(0) =0?y(0) =
1
133cos(0) + 2sin(0) + 3e
23
0+
2
13
n=1
U(0n)3cos(0n) + 2sin(0n) + 3e 23 (0n)
= 0
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
-
8/11/2019 Laplace Transform Periodic
74/84
logo1Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
-
8/11/2019 Laplace Transform Periodic
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logo1
+ 2
13
n=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
2
-
8/11/2019 Laplace Transform Periodic
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logo1
+ 2
13
n=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
3y+ 2y =
1
13
9sin(t) + 6cos(t)6e 23 t
+
2
13
n=1
U(tn)
9sin(tn) + 6cos(tn)6e 23 (tn)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
2
-
8/11/2019 Laplace Transform Periodic
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logo1
+ 2
13
n
=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
3y+ 2y =
1
13
9sin(t) + 6cos(t)6e 23 t
+
2
13
n=1
U(tn)
9sin(tn) + 6cos(tn)6e 23 (tn)
+ 113
6cos(t) + 4sin(t) + 6e 23 t+
2
13
n=1
U(tn)6cos(tn) + 4sin(tn) + 6e 23 (tn)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
2
-
8/11/2019 Laplace Transform Periodic
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logo1
+ 2
13
n
=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
3y+ 2y =
1
13
9sin(t) + 6cos(t)6e 23 t
+
2
13
n=1
U(tn)
9sin(tn) + 6cos(tn)6e 23 (tn)
+ 113
6cos(t) + 4sin(t) + 6e 23 t+
2
13
n=1
U(tn)6cos(tn) + 4sin(tn) + 6e 23 (tn)
=
1
13[9sin(t) + 4sin(t)] +
2
13
n=1U
(tn) [9sin(tn) + 4sin(tn)]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
2 2
-
8/11/2019 Laplace Transform Periodic
79/84
logo1
+ 2
13n=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
3y+ 2y =
1
13
9sin(t) + 6cos(t)6e 23 t
+
2
13
n=1
U(tn)
9sin(tn) + 6cos(tn)6e 23 (tn)
+ 113
6cos(t) + 4sin(t) + 6e 23 t+
2
13
n=1
U(tn)6cos(tn) + 4sin(tn) + 6e 23 (tn)
=
1
13[9sin(t) + 4sin(t)] +
2
13
n=1U
(tn) [9sin(tn) + 4sin(tn)]
= sin(t) + 2
n=1
U(tn) sin(tn)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
2 2
-
8/11/2019 Laplace Transform Periodic
80/84
logo1
+ 2
13n=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
3y+ 2y =
1
13
9sin(t) + 6cos(t)6e 23 t
+
2
13
n=1
U(tn)
9sin(tn) + 6cos(tn)6e 23 (tn)
+ 113
6cos(t) + 4sin(t) + 6e 23 t+
2
13
n=1
U(tn)6cos(tn) + 4sin(tn) + 6e 23 (tn)
=
1
13[9sin(t) + 4sin(t)] +
2
13
n=1U
(tn) [9sin(tn) + 4sin(tn)]
= sin(t) + 2
n=1
U(tn) sin(tn) =sin(t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
y = 1
13
3sin(t) + 2cos(t)2e 23 t
2 2 ( )
-
8/11/2019 Laplace Transform Periodic
81/84
logo1
+ 2
13n=1
U(tn)3sin(tn) + 2cos(tn)2e 2
3(tn)
3y+ 2y =
1
13
9sin(t) + 6cos(t)6e 23 t
+
2
13
n=1
U(tn)
9sin(tn) + 6cos(tn)6e 23 (tn)
+ 113
6cos(t) + 4sin(t) + 6e 23 t+
2
13
n=1
U(tn)6cos(tn) + 4sin(tn) + 6e 23 (tn)
=
1
13[9sin(t) + 4sin(t)] +
2
13
n=1U
(tn) [9sin(tn) + 4sin(tn)]
= sin(t) + 2
n=1
U(tn) sin(tn) =sin(t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Comparing Output to Input
-
8/11/2019 Laplace Transform Periodic
82/84
logo1Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Comparing Output to Input
-
8/11/2019 Laplace Transform Periodic
83/84
logo1Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
Transforms and New Formulas An Example Double Check Visualization
Comparing Output to Input
-
8/11/2019 Laplace Transform Periodic
84/84
logo1Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
