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logo1 Transforms and New Formulas An Example Double Check Laplace Transforms and Integral Equations Bernd Schr ¨ oder Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Integral Equations

Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

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Page 1: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Laplace Transforms and IntegralEquations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 2: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

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 and Integral Equations

Page 3: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo 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 and Integral Equations

Page 4: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo 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 and Integral Equations

Page 5: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t)

OriginalDE & IVP

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 6: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t)

OriginalDE & IVP

-L

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 7: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

-L

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 8: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

-L

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 9: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

-L

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 10: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Laplace transformof the solution

-L

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 11: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Laplace transformof the solution

-

L

L −1

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 12: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.

Time Domain (t) Transform domain (s)

OriginalDE & IVP

Algebraic equation forthe Laplace transform

Laplace transformof the solutionSolution

-

L

L −1

Algebraic solution,partial fractions

?

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 13: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral

1. Definite integrals of the form∫ t

0f (τ)dτ arise in circuit

theory: The charge of a capacitor is the integral of thecurrent over time.(We assume the capacitor is initially uncharged.)

2. L

{∫ t

0f (τ)dτ

}=

F(s)s

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 14: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral1. Definite integrals of the form

∫ t

0f (τ)dτ arise in circuit

theory: The charge of a capacitor is the integral of thecurrent over time.

(We assume the capacitor is initially uncharged.)

2. L

{∫ t

0f (τ)dτ

}=

F(s)s

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 15: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral1. Definite integrals of the form

∫ t

0f (τ)dτ arise in circuit

theory: The charge of a capacitor is the integral of thecurrent over time.(We assume the capacitor is initially uncharged.)

2. L

{∫ t

0f (τ)dτ

}=

F(s)s

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 16: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral1. Definite integrals of the form

∫ t

0f (τ)dτ arise in circuit

theory: The charge of a capacitor is the integral of thecurrent over time.(We assume the capacitor is initially uncharged.)

2. L

{∫ t

0f (τ)dτ

}=

F(s)s

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 17: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

sF +F−2Fs

=1s2

F(

s+1− 2s

)=

1s2

Fs2 + s−2

s=

1s2

F =s

s2(s−1)(s+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 18: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

sF +F−2Fs

=1s2

F(

s+1− 2s

)=

1s2

Fs2 + s−2

s=

1s2

F =s

s2(s−1)(s+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 19: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

sF +F−2Fs

=1s2

F(

s+1− 2s

)=

1s2

Fs2 + s−2

s=

1s2

F =s

s2(s−1)(s+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 20: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

sF +F−2Fs

=1s2

F(

s+1− 2s

)=

1s2

Fs2 + s−2

s=

1s2

F =s

s2(s−1)(s+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 21: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

sF +F−2Fs

=1s2

F(

s+1− 2s

)=

1s2

Fs2 + s−2

s=

1s2

F =s

s2(s−1)(s+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 22: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

sF +F−2Fs

=1s2

F(

s+1− 2s

)=

1s2

Fs2 + s−2

s=

1s2

F =s

s2(s−1)(s+2)Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 23: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 24: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 25: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)

s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 26: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2

, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 27: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 28: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3

, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 29: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 30: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3)

, D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 31: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 32: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2)

, A = −12

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 33: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 34: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 35: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

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Transforms and New Formulas An Example Double Check

Solve f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0

F =s

s2(s−1)(s+2)=

As

+Bs2 +

Cs−1

+D

s+2

s = As(s−1)(s+2)+B(s−1)(s+2)+Cs2(s+2)+Ds2(s−1)s = 0 : 0 = B · (−1) ·2, B = 0

s = 1 : 1 = C ·12 ·3, C =13

s = −2 : −2 = D · (−2)2 · (−3), D =16

s = −1 : −1=A·(−1)·(−2)·1+0+13·(−1)2·1+

16·(−1)2·(−2), A = −1

2

F = −12

1s

+0 · 1s2 +

13

1s−1

+16

1s+2

f (t) = −12

+13

et +16

e−2t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 36: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 37: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value:

Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 38: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f !

[−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 39: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 40: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 41: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 42: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)

+ e−2t(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 43: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)

+ t +(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 44: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t

+(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 45: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)

= t√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 46: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

Page 47: Laplace Transforms and Integral Equations · Laplace Transforms and Integral Equations. logo1 Transforms and New Formulas An Example Double Check The Laplace Transform of an Integral

logo1

Transforms and New Formulas An Example Double Check

Does f (t) = −12

+13

et +16

e−2t Really Solve the Initial Value Problem

f ′(t)+ f (t)−2∫ t

0f (z) dz = t, f (0) = 0?

Initial value: Look at f ![−1

2+

13

et +16

e−2t]′

+[−1

2+

13

et +16

e−2t]−2

∫ t

0−1

2+

13

ez +16

e−2z dz

=13

et − 13

e−2t − 12

+13

et +16

e−2t −2[−1

2z+

13

ez− 112

e−2z]t

0

=13

et − 13

e−2t − 12

+13

et +16

e−2t + t−0− 23

et +23

+16

e−2t − 16

= et(

13

+13− 2

3

)+ e−2t

(−1

3+

16

+16

)+ t +

(−1

2+

23− 1

6

)= t

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations