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Linear First Order Differential Equations Bernd Schr oder logo1 - - PowerPoint PPT Presentation

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Overview An Example Double Check Linear First Order Differential Equations Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations Overview An

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SLIDE 1

logo1 Overview An Example Double Check

Linear First Order Differential Equations

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 2

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-3
SLIDE 3

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 4

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-5
SLIDE 5

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

  • 3. To solve a linear first order differential equation, we turn

the left side into the derivative of a product.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-6
SLIDE 6

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

  • 3. To solve a linear first order differential equation, we turn

the left side into the derivative of a product.

3.1 Compute the integrating factor µ(x) = e

p(x) dx,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-7
SLIDE 7

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

  • 3. To solve a linear first order differential equation, we turn

the left side into the derivative of a product.

3.1 Compute the integrating factor µ(x) = e

p(x) dx,

3.2 Multiply the equation by the integrating factor,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-8
SLIDE 8

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

  • 3. To solve a linear first order differential equation, we turn

the left side into the derivative of a product.

3.1 Compute the integrating factor µ(x) = e

p(x) dx,

3.2 Multiply the equation by the integrating factor, note that the left side e

p(x) dxy′ +e p(x) dxp(x)y

is the derivative of the product e

p(x) dxy,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-9
SLIDE 9

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

  • 3. To solve a linear first order differential equation, we turn

the left side into the derivative of a product.

3.1 Compute the integrating factor µ(x) = e

p(x) dx,

3.2 Multiply the equation by the integrating factor, note that the left side e

p(x) dxy′ +e p(x) dxp(x)y

is the derivative of the product e

p(x) dxy,

3.3 Integrate both sides, solve for y.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-10
SLIDE 10

logo1 Overview An Example Double Check

What are Linear First Order Differential Equations?

  • 1. A linear first order differential equation is of the form

y′ +p(x)y = q(x).

  • 2. Recognizing linear first order differential equations

requires some pattern recognition.

  • 3. To solve a linear first order differential equation, we turn

the left side into the derivative of a product.

3.1 Compute the integrating factor µ(x) = e

p(x) dx,

3.2 Multiply the equation by the integrating factor, note that the left side e

p(x) dxy′ +e p(x) dxp(x)y

is the derivative of the product e

p(x) dxy,

3.3 Integrate both sides, solve for y.

That’s it.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 11

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 12

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-13
SLIDE 13

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 14

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 15

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 16

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), dx = du cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 17

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-18
SLIDE 18

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-19
SLIDE 19

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u|+c

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

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SLIDE 20

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u|

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-21
SLIDE 21

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u| = ln

  • sin(x)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-22
SLIDE 22

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u| = ln

  • sin(x)
  • µ(x)

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-23
SLIDE 23

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u| = ln

  • sin(x)
  • µ(x)

= e

p(x) dx

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-24
SLIDE 24

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u| = ln

  • sin(x)
  • µ(x)

= e

p(x) dx = eln

  • sin(x)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-25
SLIDE 25

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Integrating factor:

  • p(x) dx

=

cos(x)

sin(x) dx u := sin(x), du dx = cos(x), =

cos(x)

u du cos(x) dx = du cos(x) =

1

u du = ln|u| = ln

  • sin(x)
  • µ(x)

= e

p(x) dx = eln

  • sin(x)
  • = sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-26
SLIDE 26

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-27
SLIDE 27

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y′ + cos(x) sin(x) y = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-28
SLIDE 28

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

µ(x)

  • y′ + cos(x)

sin(x) y

  • =

1· µ(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-29
SLIDE 29

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

sin(x)

  • y′ + cos(x)

sin(x) y

  • =

1·sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-30
SLIDE 30

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

sin(x)

  • y′ + cos(x)

sin(x) y

  • =

1·sin(x) sin(x)y′ +cos(x)y = sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-31
SLIDE 31

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

sin(x)

  • y′ + cos(x)

sin(x) y

  • =

1·sin(x) sin(x)y′ +cos(x)y = sin(x) d dx

  • sin(x)y
  • =

sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-32
SLIDE 32

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

sin(x)

  • y′ + cos(x)

sin(x) y

  • =

1·sin(x) sin(x)y′ +cos(x)y = sin(x) d dx

  • sin(x)y
  • =

sin(x) sin(x)y = −cos(x)+c

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-33
SLIDE 33

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

sin(x)

  • y′ + cos(x)

sin(x) y

  • =

1·sin(x) sin(x)y′ +cos(x)y = sin(x) d dx

  • sin(x)y
  • =

sin(x) sin(x)y = −cos(x)+c y = −cos(x) sin(x) + c sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-34
SLIDE 34

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

sin(x)

  • y′ + cos(x)

sin(x) y

  • =

1·sin(x) sin(x)y′ +cos(x)y = sin(x) d dx

  • sin(x)y
  • =

sin(x) sin(x)y = −cos(x)+c y = −cos(x) sin(x) + c sin(x) = −cot(x)+ c sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-35
SLIDE 35

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-36
SLIDE 36

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-37
SLIDE 37

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x) = y(1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-38
SLIDE 38

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x) = y(1) = −cot(1)+ c sin(1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-39
SLIDE 39

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x) = y(1) = −cot(1)+ c sin(1) c sin(1) = cot(1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-40
SLIDE 40

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x) = y(1) = −cot(1)+ c sin(1) c sin(1) = cot(1) c = cot(1)sin(1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-41
SLIDE 41

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x) = y(1) = −cot(1)+ c sin(1) c sin(1) = cot(1) c = cot(1)sin(1) = cos(1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-42
SLIDE 42

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ c sin(x) = y(1) = −cot(1)+ c sin(1) c sin(1) = cot(1) c = cot(1)sin(1) = cos(1) y(x) = −cot(x)+ cos(1) sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-43
SLIDE 43

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0.

y(x) = −cot(x)+ cos(1) sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-44
SLIDE 44

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-45
SLIDE 45

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

y′ + cos(x) sin(x) y

?

= 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-46
SLIDE 46

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-47
SLIDE 47

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-48
SLIDE 48

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-49
SLIDE 49

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x) − cos2(x) sin2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-50
SLIDE 50

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x) − cos2(x) sin2(x) + cos(1)cos(x) sin2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-51
SLIDE 51

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x) − cos2(x) sin2(x) + cos(1)cos(x) sin2(x)

?

= 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-52
SLIDE 52

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x) − cos2(x) sin2(x) + cos(1)cos(x) sin2(x)

?

= 1 1−cos2(x) sin2(x)

?

= 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-53
SLIDE 53

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x) − cos2(x) sin2(x) + cos(1)cos(x) sin2(x)

?

= 1 1−cos2(x) sin2(x)

?

= 1 sin2(x) sin2(x)

?

= 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-54
SLIDE 54

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

d dx

  • −cot(x)+ cos(1)

sin(x)

  • + cos(x)

sin(x)

  • −cot(x)+ cos(1)

sin(x)

  • ?

= 1 1 sin2(x) − cos(1)cos(x) sin2(x) − cos2(x) sin2(x) + cos(1)cos(x) sin2(x)

?

= 1 1−cos2(x) sin2(x)

?

= 1 sin2(x) sin2(x) = 1 √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-55
SLIDE 55

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-56
SLIDE 56

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

y(1) =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-57
SLIDE 57

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

y(1) = −cot(1)+ cos(1) sin(1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-58
SLIDE 58

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

y(1) = −cot(1)+ cos(1) sin(1) =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-59
SLIDE 59

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

y(1) = −cot(1)+ cos(1) sin(1) = √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations

slide-60
SLIDE 60

logo1 Overview An Example Double Check

Does y(x) = −cot(x)+ cos(1) sin(x) Really Solve the Initial Value Problem y′ + cos(x) sin(x) y = 1, y(1) = 0?

y(1) = −cot(1)+ cos(1) sin(1) = √ Yes, it does

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Linear First Order Differential Equations