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Homogeneous First Order Equations Bernd Schr oder logo1 Bernd - - PowerPoint PPT Presentation

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

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SLIDE 1

logo1 Overview An Example Double Check

Homogeneous First Order Equations

Bernd Schr¨

  • der

Bernd Schr¨

  • der

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

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SLIDE 2

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

Bernd Schr¨

  • der

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

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SLIDE 3

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .

Bernd Schr¨

  • der

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

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SLIDE 4

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .
  • 2. Recognizing homogeneous first order equations requires

some pattern recognition.

Bernd Schr¨

  • der

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

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SLIDE 5

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .
  • 2. Recognizing homogeneous first order equations requires

some pattern recognition.

  • 3. To solve a homogeneous first order equation, we translate

the equation into a separable equation.

Bernd Schr¨

  • der

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

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SLIDE 6

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .
  • 2. Recognizing homogeneous first order equations requires

some pattern recognition.

  • 3. To solve a homogeneous first order equation, we translate

the equation into a separable equation.

3.1 The substitution v = y x turns the homogeneous first order equation y′ = f y x

  • into a separable equation for v,

Bernd Schr¨

  • der

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

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SLIDE 7

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .
  • 2. Recognizing homogeneous first order equations requires

some pattern recognition.

  • 3. To solve a homogeneous first order equation, we translate

the equation into a separable equation.

3.1 The substitution v = y x turns the homogeneous first order equation y′ = f y x

  • into a separable equation for v,

3.2 We can even state the resulting separable equation, but it is simpler to remember the substitution v = y x,

Bernd Schr¨

  • der

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

slide-8
SLIDE 8

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .
  • 2. Recognizing homogeneous first order equations requires

some pattern recognition.

  • 3. To solve a homogeneous first order equation, we translate

the equation into a separable equation.

3.1 The substitution v = y x turns the homogeneous first order equation y′ = f y x

  • into a separable equation for v,

3.2 We can even state the resulting separable equation, but it is simpler to remember the substitution v = y x, 3.3 After we solve the equation for v, we obtain y as xv.

Bernd Schr¨

  • der

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

slide-9
SLIDE 9

logo1 Overview An Example Double Check

What are Homogeneous First Order Equations?

  • 1. A homogeneous first order equation is of the form

y′ = f y x

  • .
  • 2. Recognizing homogeneous first order equations requires

some pattern recognition.

  • 3. To solve a homogeneous first order equation, we translate

the equation into a separable equation.

3.1 The substitution v = y x turns the homogeneous first order equation y′ = f y x

  • into a separable equation for v,

3.2 We can even state the resulting separable equation, but it is simpler to remember the substitution v = y x, 3.3 After we solve the equation for v, we obtain y as xv.

That’s it.

Bernd Schr¨

  • der

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

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SLIDE 10

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Bernd Schr¨

  • der

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

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SLIDE 11

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation.

Bernd Schr¨

  • der

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

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SLIDE 12

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x,

Bernd Schr¨

  • der

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

slide-13
SLIDE 13

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!)

Bernd Schr¨

  • der

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

slide-14
SLIDE 14

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v

Bernd Schr¨

  • der

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

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SLIDE 15

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v y′ = y x −e

y x

Bernd Schr¨

  • der

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

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SLIDE 16

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v v′x+v = y x −e

y x

Bernd Schr¨

  • der

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

slide-17
SLIDE 17

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v v′x+v = v−e

y x

Bernd Schr¨

  • der

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

slide-18
SLIDE 18

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v v′x+v = v−ev

Bernd Schr¨

  • der

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

slide-19
SLIDE 19

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v v′x+v = v−ev v′x = −ev

Bernd Schr¨

  • der

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

slide-20
SLIDE 20

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Reduction to a separable equation. v := y x, y = vx (!!) y′ = v′x+v v′x+v = v−ev v′x = −ev v′ = −1 xev

Bernd Schr¨

  • der

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

slide-21
SLIDE 21

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

Bernd Schr¨

  • der

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

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SLIDE 22

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev e−v dv = −1 x dx

Bernd Schr¨

  • der

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

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SLIDE 23

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx

Bernd Schr¨

  • der

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

slide-24
SLIDE 24

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx −e−v = −ln|x|+c

Bernd Schr¨

  • der

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

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SLIDE 25

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx −e−v = −ln|x|+c e−v = ln|x|−c

Bernd Schr¨

  • der

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

slide-26
SLIDE 26

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx −e−v = −ln|x|+c e−v = ln|x|−c −v = ln

  • ln|x|−c
  • Bernd Schr¨
  • der

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

slide-27
SLIDE 27

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx −e−v = −ln|x|+c e−v = ln|x|−c −v = ln

  • ln|x|−c
  • v

= −ln

  • ln|x|−c
  • Bernd Schr¨
  • der

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

slide-28
SLIDE 28

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx −e−v = −ln|x|+c e−v = ln|x|−c −v = ln

  • ln|x|−c
  • v

= −ln

  • ln|x|−c
  • y

= xv

Bernd Schr¨

  • der

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

slide-29
SLIDE 29

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Solving the separable equation. dv dx = −1 xev

  • e−v dv

=

  • −1

x dx −e−v = −ln|x|+c e−v = ln|x|−c −v = ln

  • ln|x|−c
  • v

= −ln

  • ln|x|−c
  • y

= xv = −xln

  • ln|x|−c
  • Bernd Schr¨
  • der

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

slide-30
SLIDE 30

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c.

Bernd Schr¨

  • der

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

slide-31
SLIDE 31

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • Bernd Schr¨
  • der

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

slide-32
SLIDE 32

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1)

Bernd Schr¨

  • der

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

slide-33
SLIDE 33

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1) = −1·ln

  • ln|1|−c
  • Bernd Schr¨
  • der

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

slide-34
SLIDE 34

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1) = −1·ln

  • ln|1|−c
  • 1

= −ln(−c)

Bernd Schr¨

  • der

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

slide-35
SLIDE 35

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1) = −1·ln

  • ln|1|−c
  • 1

= −ln(−c) ln(−c) = −1

Bernd Schr¨

  • der

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

slide-36
SLIDE 36

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1) = −1·ln

  • ln|1|−c
  • 1

= −ln(−c) ln(−c) = −1 −c = e−1

Bernd Schr¨

  • der

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

slide-37
SLIDE 37

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1) = −1·ln

  • ln|1|−c
  • 1

= −ln(−c) ln(−c) = −1 −c = e−1 c = −e−1

Bernd Schr¨

  • der

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

slide-38
SLIDE 38

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

Finding c. y = −xln

  • ln|x|−c
  • 1

= y(1) = −1·ln

  • ln|1|−c
  • 1

= −ln(−c) ln(−c) = −1 −c = e−1 c = −e−1 y = −xln

  • ln|x|+ 1

e

  • Bernd Schr¨
  • der

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

slide-39
SLIDE 39

logo1 Overview An Example Double Check

Solve the Initial Value Problem y′ = y x −e

y x,

y(1) = 1.

y = −xln

  • ln|x|+ 1

e

  • Bernd Schr¨
  • der

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

slide-40
SLIDE 40

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ =

Bernd Schr¨

  • der

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

slide-41
SLIDE 41

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ = −ln

  • ln|x|+ 1

e

  • +(−x)

1 ln|x|+ 1

e

1 x

Bernd Schr¨

  • der

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

slide-42
SLIDE 42

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ = −ln

  • ln|x|+ 1

e

  • +(−x)

1 ln|x|+ 1

e

1 x = −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

Bernd Schr¨

  • der

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

slide-43
SLIDE 43

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ = −ln

  • ln|x|+ 1

e

  • +(−x)

1 ln|x|+ 1

e

1 x = −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

y x −e

y x

=

Bernd Schr¨

  • der

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

slide-44
SLIDE 44

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ = −ln

  • ln|x|+ 1

e

  • +(−x)

1 ln|x|+ 1

e

1 x = −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

y x −e

y x

= −ln

  • ln|x|+ 1

e

  • −e−ln(ln|x|+ 1

e)

Bernd Schr¨

  • der

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

slide-45
SLIDE 45

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ = −ln

  • ln|x|+ 1

e

  • +(−x)

1 ln|x|+ 1

e

1 x = −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

y x −e

y x

= −ln

  • ln|x|+ 1

e

  • −e−ln(ln|x|+ 1

e)

= −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

Bernd Schr¨

  • der

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

slide-46
SLIDE 46

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y′ = −ln

  • ln|x|+ 1

e

  • +(−x)

1 ln|x|+ 1

e

1 x = −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

y x −e

y x

= −ln

  • ln|x|+ 1

e

  • −e−ln(ln|x|+ 1

e)

= −ln

  • ln|x|+ 1

e

  • −

1 ln|x|+ 1

e

√

Bernd Schr¨

  • der

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

slide-47
SLIDE 47

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) =

Bernd Schr¨

  • der

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

slide-48
SLIDE 48

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) = −1·ln

  • ln|1|+ 1

e

  • Bernd Schr¨
  • der

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

slide-49
SLIDE 49

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) = −1·ln

  • ln|1|+ 1

e

  • =

−ln 1 e

  • Bernd Schr¨
  • der

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

slide-50
SLIDE 50

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) = −1·ln

  • ln|1|+ 1

e

  • =

−ln 1 e

  • =

−(−1)

Bernd Schr¨

  • der

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

slide-51
SLIDE 51

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) = −1·ln

  • ln|1|+ 1

e

  • =

−ln 1 e

  • =

−(−1) = 1

Bernd Schr¨

  • der

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

slide-52
SLIDE 52

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) = −1·ln

  • ln|1|+ 1

e

  • =

−ln 1 e

  • =

−(−1) = 1 √

Bernd Schr¨

  • der

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

slide-53
SLIDE 53

logo1 Overview An Example Double Check

Does y = −xln

  • ln|x|+ 1

e

  • Really Solve the

Initial Value Problem y′ = y x −e

y x, y(1) = 1?

y(1) = −1·ln

  • ln|1|+ 1

e

  • =

−ln 1 e

  • =

−(−1) = 1 √ Yes, it does.

Bernd Schr¨

  • der

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