Laplace Transforms and Integral Equations Laplace Transforms and Integral Equations. logo1 Transforms

logo1

Transforms and New Formulas An Example Double Check

Laplace Transforms and Integral Equations

Bernd Schröder

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 of the solution

-L

Algebraic solution, partial fractions

?

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 of the solution

-

�

L

L −1

Algebraic solution, partial fractions

?

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

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 of the solutionSolution

-

�

L

L −1

Algebraic solution, partial fractions

?

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral

1. Definite integrals of the form ∫ t

0 f (τ)dτ arise in circuit

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

2. L {∫ t

0 f (τ)dτ

} =

F(s) s

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral 1. Definite integrals of the form

∫ t 0

f (τ)dτ arise in circuit theory: The charge of a capacitor is the integral of the current over time.

(We assume the capacitor is initially uncharged.)

2. L {∫ t

0 f (τ)dτ

} =

F(s) s

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral 1. Definite integrals of the form

∫ t 0

f (τ)dτ arise in circuit theory: The charge of a capacitor is the integral of the current over time. (We assume the capacitor is initially uncharged.)

2. L {∫ t

0 f (τ)dτ

} =

F(s) s

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

The Laplace Transform of an Integral 1. Definite integrals of the form

∫ t 0

f (τ)dτ arise in circuit theory: The charge of a capacitor is the integral of the current over time. (We assume the capacitor is initially uncharged.)

2. L {∫ t

0 f (τ)dτ

} =

F(s) s

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2 ∫ t

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

f ′(t)+ f (t)−2 ∫ t

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

sF +F−2F s

= 1 s2

F (

s+1− 2 s

) =

1 s2

F s2 + s−2

s =

1 s2

F = s

s2(s−1)(s+2)

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2 ∫ t

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

f ′(t)+ f (t)−2 ∫ t

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

sF +F−2F s

= 1 s2

F (

s+1− 2 s

) =

1 s2

F s2 + s−2

s =

1 s2

F = s

s2(s−1)(s+2)

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

Solve the Initial Value Problem

f ′(t)+ f (t)−2 ∫ t

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

f ′(t)+ f (t)−2 ∫ t

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

sF +F−2F s

= 1 s2

F (

s+1− 2 s

) =

1 s2

F s2 + s−2

s =

1 s2

F = s

s2(s−1)(s+2)

Bernd Schröder Louisiana Tech University, College of Engineering and Science

Laplace Transforms and Integral Equations

logo1

Transforms and New Formulas An Example Double Check

So