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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