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The Method of Undetermined Coefficients for Forcing Functions that - - PowerPoint PPT Presentation

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Overview An Example Double Check Further Discussion The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of

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SLIDE 1

logo1 Overview An Example Double Check Further Discussion

The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 2

logo1 Overview An Example Double Check Further Discussion

The Method of Undetermined Coefficients

  • 1. The general solution of an inhomogeneous linear

differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 3

logo1 Overview An Example Double Check Further Discussion

The Method of Undetermined Coefficients

  • 1. The general solution of an inhomogeneous linear

differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp +yh

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 4

logo1 Overview An Example Double Check Further Discussion

The Method of Undetermined Coefficients

  • 1. The general solution of an inhomogeneous linear

differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp +yh

  • 2. In the Method of Undetermined Coefficients we detect

repeating patterns in the derivatives of the inhomogeneity and set up the particular solution as a linear combination of the patterns with undetermined coefficients.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 5

logo1 Overview An Example Double Check Further Discussion

The Method of Undetermined Coefficients

  • 1. The general solution of an inhomogeneous linear

differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp +yh

  • 2. In the Method of Undetermined Coefficients we detect

repeating patterns in the derivatives of the inhomogeneity and set up the particular solution as a linear combination of the patterns with undetermined coefficients.

  • 3. The conjectured solution is substituted into the equation to

determine the coefficients.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 6

logo1 Overview An Example Double Check Further Discussion

The Method of Undetermined Coefficients

  • 1. The general solution of an inhomogeneous linear

differential equation is the sum of a particular solution of the inhomogeneous equation and the general solution of the corresponding homogeneous equation. y = yp +yh

  • 2. In the Method of Undetermined Coefficients we detect

repeating patterns in the derivatives of the inhomogeneity and set up the particular solution as a linear combination of the patterns with undetermined coefficients.

  • 3. The conjectured solution is substituted into the equation to

determine the coefficients.

  • 4. When the forcing function is a solution of the

homogeneous equation, multiply it with the independent variable until it no longer solves the homogeneous

  • equation. Start your pattern with that function.

That’s it.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 7

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 8

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 9

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 10

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y = λ 2eλx +9eλx =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 11

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y = λ 2eλx +9eλx = λ 2 +9 =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 12

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y = λ 2eλx +9eλx = λ 2 +9 = λ 2 = −9

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 13

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y = λ 2eλx +9eλx = λ 2 +9 = λ 2 = −9 λ1,2 = ±3i

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 14

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y = λ 2eλx +9eλx = λ 2 +9 = λ 2 = −9 λ1,2 = ±3i yh = c1 cos(3x)+c2 sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 15

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Solution of the homogeneous equation. y′′ +9y = λ 2eλx +9eλx = λ 2 +9 = λ 2 = −9 λ1,2 = ±3i yh = c1 cos(3x)+c2 sin(3x) Need to multiply the right side by x to get the function that starts the pattern.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 16

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 17

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 18

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 19

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 20

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 21

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 22

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x)) (no need for a term cos(3x))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 23

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x)) (no need for a term cos(3x)) r′′(x) = −9xsin(3x)+3cos(3x)+3cos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 24

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x)) (no need for a term cos(3x)) r′′(x) = −9xsin(3x)+3cos(3x)+3cos(3x) (already have xsin(3x))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 25

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x)) (no need for a term cos(3x)) r′′(x) = −9xsin(3x)+3cos(3x)+3cos(3x) (already have xsin(3x)) (no need for terms cos(3x),sin(3x))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 26

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x)) (no need for a term cos(3x)) r′′(x) = −9xsin(3x)+3cos(3x)+3cos(3x) (already have xsin(3x)) (no need for terms cos(3x),sin(3x)) (after this, it “repeats”)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 27

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Generating the form of the particular solution of the inhomogeneous equation. r(x) = xsin(3x) (need a term xsin(3x)) r′(x) = 3xcos(3x)+sin(3x) (need a term xcos(3x)) (no need for a term cos(3x)) r′′(x) = −9xsin(3x)+3cos(3x)+3cos(3x) (already have xsin(3x)) (no need for terms cos(3x),sin(3x)) (after this, it “repeats”) yp := Axcos(3x)+Bxsin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 28

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Preparing yp.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 29

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Preparing yp. yp = Axcos(3x)+Bxsin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 30

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Preparing yp. yp = Axcos(3x)+Bxsin(3x) y′

p

= −3Axsin(3x)+Acos(3x)+3Bxcos(3x)+Bsin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 31

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Preparing yp. yp = Axcos(3x)+Bxsin(3x) y′

p

= −3Axsin(3x)+Acos(3x)+3Bxcos(3x)+Bsin(3x) y′′

p

= −9Axcos(3x)−3Asin(3x)−3Asin(3x) −9Bxsin(3x)+3Bcos(3x)+3Bcos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 32

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Preparing yp. yp = Axcos(3x)+Bxsin(3x) y′

p

= −3Axsin(3x)+Acos(3x)+3Bxcos(3x)+Bsin(3x) y′′

p

= −9Axcos(3x)−3Asin(3x)−3Asin(3x) −9Bxsin(3x)+3Bcos(3x)+3Bcos(3x) = −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 33

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 34

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

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SLIDE 35

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-36
SLIDE 36

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-37
SLIDE 37

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x) 6B = 0,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-38
SLIDE 38

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x) 6B = 0, B = 0,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-39
SLIDE 39

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x) 6B = 0, B = 0, −6A = 1,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-40
SLIDE 40

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x) 6B = 0, B = 0, −6A = 1, A = −1 6

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-41
SLIDE 41

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x) 6B = 0, B = 0, −6A = 1, A = −1 6 yp = −1 6xcos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-42
SLIDE 42

logo1 Overview An Example Double Check Further Discussion

Solve the Differential Equation y′′ +9y = sin(3x)

Determining the coefficients. y′′ +9y = sin(3x)

  • −9Axcos(3x)−6Asin(3x)−9Bxsin(3x)+6Bcos(3x)
  • +9
  • Axcos(3x)+Bxsin(3x)
  • =

sin(3x) (−9A+9A)xcos(3x)+(−9B+9B)xsin(3x) +(6B)cos(3x)+(−6A)sin(3x) = sin(3x) 6B = 0, B = 0, −6A = 1, A = −1 6 yp = −1 6xcos(3x) y = −1 6xcos(3x)+c1 cos(3x)+c2 sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-43
SLIDE 43

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-44
SLIDE 44

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-45
SLIDE 45

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-46
SLIDE 46

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-47
SLIDE 47

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x) = 3 2xcos(3x)+sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-48
SLIDE 48

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x) = 3 2xcos(3x)+sin(3x) y′′ +9y

?

= sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-49
SLIDE 49

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x) = 3 2xcos(3x)+sin(3x) y′′ +9y

?

= sin(3x) 3 2xcos(3x)+sin(3x)+9

  • −1

6xcos(3x)

  • ?

= sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-50
SLIDE 50

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x) = 3 2xcos(3x)+sin(3x) y′′ +9y

?

= sin(3x) 3 2xcos(3x)+sin(3x)+9

  • −1

6xcos(3x)

  • ?

= sin(3x) 3 2xcos(3x)+sin(3x)− 3 2xcos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-51
SLIDE 51

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x) = 3 2xcos(3x)+sin(3x) y′′ +9y

?

= sin(3x) 3 2xcos(3x)+sin(3x)+9

  • −1

6xcos(3x)

  • ?

= sin(3x) 3 2xcos(3x)+sin(3x)− 3 2xcos(3x) = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-52
SLIDE 52

logo1 Overview An Example Double Check Further Discussion

Does yp = −1 6xcos(3x) Really Solve the Differential Equation y′′ +9y = sin(3x)

yp = −1 6xcos(3x), y′

p = 1

2xsin(3x)− 1 6 cos(3x), y′′

p = 3

2xcos(3x)+ 1 2 sin(3x)+ 1 2 sin(3x) = 3 2xcos(3x)+sin(3x) y′′ +9y

?

= sin(3x) 3 2xcos(3x)+sin(3x)+9

  • −1

6xcos(3x)

  • ?

= sin(3x) 3 2xcos(3x)+sin(3x)− 3 2xcos(3x) = sin(3x) √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-53
SLIDE 53

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-54
SLIDE 54

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x),

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-55
SLIDE 55

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x), y′′ = −9Asin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-56
SLIDE 56

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x), y′′ = −9Asin(3x) y′′ +9y = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-57
SLIDE 57

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x), y′′ = −9Asin(3x) y′′ +9y = sin(3x) −9Asin(3x)+9Asin(3x) = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-58
SLIDE 58

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x), y′′ = −9Asin(3x) y′′ +9y = sin(3x) −9Asin(3x)+9Asin(3x) = sin(3x) = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-59
SLIDE 59

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x), y′′ = −9Asin(3x) y′′ +9y = sin(3x) −9Asin(3x)+9Asin(3x) = sin(3x) = sin(3x) ???

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-60
SLIDE 60

logo1 Overview An Example Double Check Further Discussion

What would have happened if we had used sin(3x) or cos(3x)?

y = Asin(3x), y′ = 3Acos(3x), y′′ = −9Asin(3x) y′′ +9y = sin(3x) −9Asin(3x)+9Asin(3x) = sin(3x) = sin(3x) ??? Whenever you set up the Method of Undetermined Coefficients and something zeros out, double check if you started with a solution of the homogeneous equation and adjust appropriately.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-61
SLIDE 61

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-62
SLIDE 62

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

  • 1. The starting term is the forcing function, adjusted by

multiplying by powers of x, as demonstrated here.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-63
SLIDE 63

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

  • 1. The starting term is the forcing function, adjusted by

multiplying by powers of x, as demonstrated here.

  • 2. Compute derivatives of the starting term until no new

patterns emerge.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-64
SLIDE 64

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

  • 1. The starting term is the forcing function, adjusted by

multiplying by powers of x, as demonstrated here.

  • 2. Compute derivatives of the starting term until no new

patterns emerge.

  • 3. Generate a term (with an undetermined coefficient) for

every new term that occurs in one of the derivatives, except for solutions of the homogeneous equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-65
SLIDE 65

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-66
SLIDE 66

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-67
SLIDE 67

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-68
SLIDE 68

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-69
SLIDE 69

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = e−x (solves the homogeneous equation)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-70
SLIDE 70

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-71
SLIDE 71

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = xe−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-72
SLIDE 72

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = xe−x (solves the homogeneous equation)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-73
SLIDE 73

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = xe−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-74
SLIDE 74

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-75
SLIDE 75

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-76
SLIDE 76

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-77
SLIDE 77

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

r′(x) = 2xe−x −x2e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-78
SLIDE 78

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

r′(x) = 2xe−x −x2e−x (no new terms)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-79
SLIDE 79

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

r′(x) = 2xe−x −x2e−x (no new terms) r′′(x) = 2e−x −4xe−x +x2e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-80
SLIDE 80

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

r′(x) = 2xe−x −x2e−x (no new terms) r′′(x) = 2e−x −4xe−x +x2e−x (no new terms)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-81
SLIDE 81

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

r′(x) = 2xe−x −x2e−x (no new terms) r′′(x) = 2e−x −4xe−x +x2e−x (no new terms) (it “repeats” from here)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-82
SLIDE 82

logo1 Overview An Example Double Check Further Discussion

How to Generate the Setup When the Forcing Function Solves the Homogeneous Equation

y′′ +2y′ +y = e−x yh = c1e−x +c2xe−x r(x) = x2e−x (does not solve the homogeneous equation)

  • need a term x2e−x

r′(x) = 2xe−x −x2e−x (no new terms) r′′(x) = 2e−x −4xe−x +x2e−x (no new terms) (it “repeats” from here) yp = Ax2e−x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-83
SLIDE 83

logo1 Overview An Example Double Check Further Discussion

Resonance

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-84
SLIDE 84

logo1 Overview An Example Double Check Further Discussion

Resonance

y′′ +9y = sin(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-85
SLIDE 85

logo1 Overview An Example Double Check Further Discussion

Resonance

y′′ +9y = sin(3x) , yp = −1 6xcos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation

slide-86
SLIDE 86

logo1 Overview An Example Double Check Further Discussion

Resonance

y′′ +9y = sin(3x) , yp = −1 6xcos(3x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Method of Undetermined Coefficients for Forcing Functions that Solve the Homogeneous Equation