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

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  • 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