Separable IVP

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    Solving Initial Value Problems An Example Double Check

    An Initial Value Problem for a Separable

    Differential Equation

    Bernd Schroder

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

  • 7/28/2019 Separable IVP

    2/69

    logo1

    Solving Initial Value Problems An Example Double Check

    Approach

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

  • 7/28/2019 Separable IVP

    3/69

    logo1

    Solving Initial Value Problems An Example Double Check

    ApproachThis approach will work as long as the general solution of the

    differential equation can be computed.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

  • 7/28/2019 Separable IVP

    4/69

    logo1

    Solving Initial Value Problems An Example Double Check

    ApproachThis approach will work as long as the general solution of the

    differential equation can be computed.

    1. Find the general solution of the differential equation.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

  • 7/28/2019 Separable IVP

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    logo1

    Solving Initial Value Problems An Example Double Check

    ApproachThis approach will work as long as the general solution of the

    differential equation can be computed.

    1. Find the general solution of the differential equation.

    2. Use the initial conditions to determine the value(s) of the

    constant(s) in the general solution.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

  • 7/28/2019 Separable IVP

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    logo1

    Solving Initial Value Problems An Example Double Check

    ApproachThis approach will work as long as the general solution of the

    differential equation can be computed.

    1. Find the general solution of the differential equation.

    2. Use the initial conditions to determine the value(s) of the

    constant(s) in the general solution.

    3. Double check if the solution works.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    S l i I iti l V l P bl A E l D bl Ch k

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    logo1

    Solving Initial Value Problems An Example Double Check

    ApproachThis approach will work as long as the general solution of the

    differential equation can be computed.

    1. Find the general solution of the differential equation.

    2. Use the initial conditions to determine the value(s) of the

    constant(s) in the general solution.

    3. Double check if the solution works.Thats it.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    S l i I iti l V l P bl A E l D bl Ch k

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    logo1

    Solving Initial Value Problems An Example Double Check

    Solve the Initial Value Problem

    y = xsin(x)y+ xsin(x)y

    , y(0) = 1.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

  • 7/28/2019 Separable IVP

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    logo1

    Solving Initial Value Problems An Example Double Check

    Solve the Initial Value Problem

    y = xsin(x)y+ xsin(x)y

    , y(0) = 1.

    y

    = xsin(x)y+xsin(x)

    y

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

  • 7/28/2019 Separable IVP

    10/69

    logo1

    Solving Initial Value Problems An Example Double Check

    Solve the Initial Value Problem

    y = xsin(x)y+ xsin(x)y

    , y(0) = 1.

    y

    = xsin(x)y+xsin(x)

    y

    y = xsin(x)y+

    1

    y

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving Initial Value Problems An Example Double Check

    Solve the Initial Value Problem

    y = xsin(x)y+ xsin(x)y

    , y(0) = 1.

    y = xsin(x)y+xsin(x)

    y

    y = xsin(x)y+

    1

    y

    dy

    dx = xsin(x)y

    2 + 1

    y

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving Initial Value Problems An Example Double Check

    Solve the Initial Value Problem

    y = xsin(x)y+ xsin(x)y

    , y(0) = 1.

    y = xsin(x)y+xsin(x)

    y

    y = xsin(x)y+

    1

    y

    dy

    dx = xsin(x)y

    2 + 1

    y

    y

    y2 + 1dy = xsin(x) dx

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    g p

    Solve the Initial Value Problem

    y = xsin(x)y+ xsin(x)y

    , y(0) = 1.

    y = xsin(x)y+xsin(x)

    y

    y = xsin(x)y+

    1

    y

    dy

    dx = xsin(x)y

    2 + 1

    y

    y

    y2 + 1dy =

    xsin(x) dx

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    g p

    The Integral

    y

    y2 + 1dy.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.du

    dy= 2y, so dy =

    du

    2y.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.du

    dy= 2y, so dy =

    du

    2y.

    y

    y2

    + 1

    dy =

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

  • 7/28/2019 Separable IVP

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.du

    dy= 2y, so dy =

    du

    2y.

    y

    y2

    + 1

    dy = y

    u

    du

    2y

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

  • 7/28/2019 Separable IVP

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.du

    dy= 2y, so dy =

    du

    2y.

    y

    y2

    + 1

    dy = y

    u

    du

    2y

    =1

    2

    1

    udu

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.du

    dy= 2y, so dy =

    du

    2y.

    y

    y2

    + 1

    dy = y

    u

    du

    2y

    =1

    2

    1

    udu

    =1

    2ln

    |u

    |+ c

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

  • 7/28/2019 Separable IVP

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    logo1

    The Integral

    y

    y2 + 1dy.

    Idea: Substitution u := y2 + 1.du

    dy= 2y, so dy =

    du

    2y.

    y

    y2

    + 1

    dy = y

    u

    du

    2y

    =1

    2

    1

    udu

    =1

    2ln

    |u

    |+ c

    =1

    2ln

    y2 + 1+ c

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    The Integral

    xsin(x) dx.

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    xsin(x) dx.

    Idea: Integration by parts.

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    xsin(x) dx =

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    xsin(x) dx = x cos(x)

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    xsin(x) dx = x cos(x)

    1 cos(x) dx

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    xsin(x) dx = x cos(x)

    1 cos(x) dx

    = xcos(x) +cos(x) dx

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    xsin(x) dx = x cos(x)

    1 cos(x) dx

    = xcos(x) +cos(x) dx= xcos(x) + sin(x) + c

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    The Integral

    xsin(x) dx.

    Idea: Integration by parts.Integrate sin(x), differentiate x.

    xsin(x) dx = x cos(x)

    1 cos(x) dx

    = xcos(x) +cos(x) dx= xcos(x) + sin(x) + c= sin(x)xcos(x) + c

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    =

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    = sin(x)xcos(x) + c

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    = sin(x)xcos(x) + c

    lny2 + 1 = 2sin(x)2xcos(x) + c

    Bernd Schroder Louisiana Tech University, College of Engineering and ScienceAn Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    = sin(x)xcos(x) + c

    lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = e2sin(x)2xcos(x)+c

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    = sin(x)xcos(x) + c

    lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = ke2sin(x)2xcos(x)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    = sin(x)xcos(x) + c

    lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = ke2sin(x)2xcos(x)

    y2 = ke2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    ... Continuing Where We Left Off.

    y

    y2 + 1dy =

    xsin(x) dx

    1

    2ln

    y2 + 1

    = sin(x)xcos(x) + c

    lny2 + 1 = 2sin(x)2xcos(x) + cy2 + 1 = ke2sin(x)2xcos(x)

    y2 = ke2sin(x)2xcos(x) 1

    y = ke2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 =

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 = ke

    2sin(0)20cos(0) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 = ke

    2sin(0)20cos(0) 11 = ke0 1 =

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 = ke

    2sin(0)20cos(0) 11 = ke0 1 = k1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 = ke

    2sin(0)20cos(0) 11 = ke0 1 = k11 =

    k1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 =

    ke2sin(0)20cos(0) 1

    1 = ke0 1 = k11 =

    k1

    1 = k1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 =

    ke2sin(0)20cos(0) 1

    1 = ke0 1 = k11 =

    k1

    1 = k1k = 2

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

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    logo1

    Solving the Initial Value Problem

    1 = y(0)

    y(x) =

    ke2sin(x)2xcos(x) 11 =

    ke2sin(0)20cos(0) 1

    1 = ke0 1 = k11 =

    k1

    1 = k1k = 2

    y(x) =

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    S l i h I i i l V l P bl

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    logo1

    Solving the Initial Value Problem

    y(x) =

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    S l i h I i i l V l P bl

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    logo1

    Solving the Initial Value Problem

    y(x) =

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    D ( )

    2 2sin( ) 2 cos( ) 1 S l h IVP

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    logo1

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    y(x) =

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    D ( )

    2 2sin(x) 2xcos(x) 1 S l th IVP

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    logo1

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    y(x) =

    2e2sin(x)2xcos(x) 1y(0) =

    2e2sin(0)20cos(0) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    D ( )

    2 2sin(x) 2xcos(x) 1 S l th IVP

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    logo1

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    y(x) =

    2e2sin(x)2xcos(x) 1y(0) =

    2e2sin(0)20cos(0) 1 = 2e0 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    D ( )

    2 2sin(x)2xcos(x) 1 S l th IVP

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    logo1

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    y(x) =

    2e2sin(x)2xcos(x) 1

    y(0) =

    2e2sin(0)20cos(0) 1 = 2e0 1 = 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does ( )

    2 2sin(x)2xcos(x) 1 Solve the IVP

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    logo1

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    y(x) =

    2e2sin(x)2xcos(x) 1

    y(0) =

    2e2sin(0)20cos(0) 1 = 2e0 1 = 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x)

    2e2sin(x)2xcos(x) 1 Solve the IVP

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    logo1

    Does y(x) =

    2e2sin(x) 2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

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    Does y(x) =

    2e2sin(x) 2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    =1

    2

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

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    logo1

    Does y(x) =

    2e2sin(x) 2xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    =1

    2

    2e2sin(x)2xcos(x) 12e2sin(x)2xcos(x)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

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    logo1

    Does y(x) =

    2e s (x) xcos(x) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    =2e2sin(x)2xcos(x)

    2

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

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    61/69

    logo1

    Does y(x) =

    2e ( ) ( ) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    =2e2sin(x)2xcos(x)

    2

    2e2sin(x)2xcos(x) 1

    2cos(x)2cos(x) + 2xsin(x)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

  • 7/28/2019 Separable IVP

    62/69

    logo1

    Does y(x) =

    2e ( ) ( ) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    =2e2sin(x)2xcos(x)

    2

    2e2sin(x)2xcos(x) 1

    2cos(x)2cos(x) + 2xsin(x)

    =2e2sin(x)2xcos(x)

    2

    2e2sin(x)2xcos(x) 12xsin(x)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    Does y(x) =

    2e2sin(x)2xcos(x) 1 Solve the IVP

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    63/69

    logo1

    Does y(x) =

    2e ( ) ( ) 1 Solve the IVP

    y = xsin(x)y+xsin(x)

    y , y(0) = 1?

    d

    dx 2e2sin(x)2xcos(x) 1

    =2e2sin(x)2xcos(x)

    2

    2e2sin(x)2xcos(x) 1

    2cos(x)2cos(x) + 2xsin(x)

    =2e2sin(x)2xcos(x)

    2

    2e2sin(x)2xcos(x) 12xsin(x)

    =2e2sin(x)2xcos(x)

    2e2sin(x)2xcos(x) 1xsin(x)

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    i ( )

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    logo1

    xsin(x)y+xsin(x)

    y

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    i ( )

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    logo1

    xsin(x)y+xsin(x)

    y

    = xsin(x)

    2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    i ( )

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    logo1

    xsin(x)y+xsin(x)

    y

    = xsin(x)

    2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1

    =

    xsin(x)2e2sin(x)2xcos(x) 1

    2

    +xsin(x)2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    i ( )

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    logo1

    xsin(x)y+xsin(x)

    y

    = xsin(x)

    2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1

    =

    xsin(x)2e2sin(x)2xcos(x) 1

    2

    +xsin(x)2e2sin(x)2xcos(x) 1

    =xsin(x)

    2e2sin(x)2xcos(x) 1 + 1

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    xsin(x)

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    logo1

    xsin(x)y+xsin(x)

    y

    = xsin(x)

    2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1

    =

    xsin(x)2e2sin(x)2xcos(x) 1

    2

    +xsin(x)2e2sin(x)2xcos(x) 1

    =xsin(x)

    2e2sin(x)2xcos(x) 1 + 1

    2e2sin(x)2xcos(x) 1

    = xsin(x)2e2sin(x)2xcos(x)

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation

    Solving Initial Value Problems An Example Double Check

    xsin(x)

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    logo1

    xsin(x)y+xsin(x)

    y

    = xsin(x)

    2e2sin(x)2xcos(x) 1 + xsin(x)2e2sin(x)2xcos(x) 1

    =

    xsin(x)

    2e2sin(x)2xcos(x) 1

    2

    +xsin(x)

    2e2sin(x)2xcos(x) 1

    =xsin(x)

    2e2sin(x)2xcos(x) 1 + 1

    2e2sin(x)2xcos(x) 1

    = xsin(x)2e2sin(x)2xcos(x)

    2e2sin(x)2xcos(x) 1

    Bernd Schroder Louisiana Tech University, College of Engineering and Science

    An Initial Value Problem for a Separable Differential Equation