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Using Laplace Transforms to Solve Initial Value · PDF fileUsing Laplace Transforms to Solve Initial Value Problems ... Algebraic equation for the Laplace transform L ... Using Laplace

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  • logo1

    Overview An Example Double Check

    Using Laplace Transforms to Solve InitialValue Problems

    Bernd Schroder

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    Time Domain (t)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    Time Domain (t)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    Time Domain (t)

    OriginalDE & IVP

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    Time Domain (t)

    OriginalDE & IVP

    -L

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    Time Domain (t)

    OriginalDE & IVP

    Algebraic equation forthe Laplace transform

    -L

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    Time Domain (t) Transform domain (s)

    OriginalDE & IVP

    Algebraic equation forthe Laplace transform

    -L

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    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

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    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

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    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

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    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

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.1.1 L {y}(s) =: Y(s) (This is just notation.)1.2 L

    {y

    }(s) = sY(s) y(0)

    1.3 L{

    y}

    (s) = s2Y(s) sy(0) y(0)1.4 L

    {y(n)(t)

    }(s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)

    2. The right sides above do not involve derivatives ofwhatever Y is.

    3. The other key property is that constants and sums factorthrough the Laplace transform:L {f +g} = L {f}+L {g} and L {af} = aL {f}.(That is, the Laplace transform is linear.)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.

    1.1 L {y}(s) =: Y(s) (This is just notation.)1.2 L

    {y

    }(s) = sY(s) y(0)

    1.3 L{

    y}

    (s) = s2Y(s) sy(0) y(0)1.4 L

    {y(n)(t)

    }(s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)

    2. The right sides above do not involve derivatives ofwhatever Y is.

    3. The other key property is that constants and sums factorthrough the Laplace transform:L {f +g} = L {f}+L {g} and L {af} = aL {f}.(That is, the Laplace transform is linear.)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.1.1 L {y}(s) =: Y(s)

    (This is just notation.)1.2 L

    {y

    }(s) = sY(s) y(0)

    1.3 L{

    y}

    (s) = s2Y(s) sy(0) y(0)1.4 L

    {y(n)(t)

    }(s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)

    2. The right sides above do not involve derivatives ofwhatever Y is.

    3. The other key property is that constants and sums factorthrough the Laplace transform:L {f +g} = L {f}+L {g} and L {af} = aL {f}.(That is, the Laplace transform is linear.)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.1.1 L {y}(s) =: Y(s) (This is just notation.)

    1.2 L{

    y}

    (s) = sY(s) y(0)1.3 L

    {y

    }(s) = s2Y(s) sy(0) y(0)

    1.4 L{

    y(n)(t)}

    (s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)2. The right sides above do not involve derivatives of

    whatever Y is.3. The other key property is that constants and sums factor

    through the Laplace transform:L {f +g} = L {f}+L {g} and L {af} = aL {f}.(That is, the Laplace transform is linear.)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.1.1 L {y}(s) =: Y(s) (This is just notation.)1.2 L

    {y

    }(s) = sY(s) y(0)

    1.3 L{

    y}

    (s) = s2Y(s) sy(0) y(0)1.4 L

    {y(n)(t)

    }(s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)

    2. The right sides above do not involve derivatives ofwhatever Y is.

    3. The other key property is that constants and sums factorthrough the Laplace transform:L {f +g} = L {f}+L {g} and L {af} = aL {f}.(That is, the Laplace transform is linear.)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.1.1 L {y}(s) =: Y(s) (This is just notation.)1.2 L

    {y

    }(s) = sY(s) y(0)

    1.3 L{

    y}

    (s) = s2Y(s) sy(0) y(0)

    1.4 L{

    y(n)(t)}

    (s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)2. The right sides above do not involve derivatives of

    whatever Y is.3. The other key property is that constants and sums factor

    through the Laplace transform:L {f +g} = L {f}+L {g} and L {af} = aL {f}.(That is, the Laplace transform is linear.)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    Using Laplace Transforms to Solve Initial Value Problems

  • logo1

    Overview An Example Double Check

    How Laplace Transforms Turn Initial ValueProblems Into Algebraic Equations

    1. The first key property of the Laplace transform is the wayderivatives are transformed.1.1 L {y}(s) =: Y(s) (This is just notation.)1.2 L

    {y

    }(s) = sY(s) y(0)

    1.3 L{

    y}

    (s) = s2Y(s) sy(0) y(0)1.4 L

    {y(n)(t)

    }(s) = snY(s) sn1y(0) sn2y(0) y(n1)(0)

    2. The right sides above do not involve derivatives ofwhateve

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