Cauchy-Euler Equations Bernd Schr oder logo1 Bernd Schr oder - - PowerPoint PPT Presentation

▶
cauchy euler equations
SMART_READER_LITE
LIVE PREVIEW

Cauchy-Euler Equations Bernd Schr oder logo1 Bernd Schr oder - - PowerPoint PPT Presentation

Dec 12, 2022 222 likes •969 views

Overview An Example Double Check Discussion Cauchy-Euler Equations Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations Overview An Example Double Check

slide-1
SLIDE 1

logo1 Overview An Example Double Check Discussion

Cauchy-Euler Equations

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-2
SLIDE 2

logo1 Overview An Example Double Check Discussion

Definition and Solution Method

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-3
SLIDE 3

logo1 Overview An Example Double Check Discussion

Definition and Solution Method

  • 1. A second order Cauchy-Euler equation is of the form

a2x2d2y dx2 +a1xdy dx +a0y = g(x).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-4
SLIDE 4

logo1 Overview An Example Double Check Discussion

Definition and Solution Method

  • 1. A second order Cauchy-Euler equation is of the form

a2x2d2y dx2 +a1xdy dx +a0y = g(x). If g(x) = 0, then the equation is called homogeneous.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-5
SLIDE 5

logo1 Overview An Example Double Check Discussion

Definition and Solution Method

  • 1. A second order Cauchy-Euler equation is of the form

a2x2d2y dx2 +a1xdy dx +a0y = g(x). If g(x) = 0, then the equation is called homogeneous.

  • 2. To solve a homogeneous Cauchy-Euler equation we set

y = xr and solve for r.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-6
SLIDE 6

logo1 Overview An Example Double Check Discussion

Definition and Solution Method

  • 1. A second order Cauchy-Euler equation is of the form

a2x2d2y dx2 +a1xdy dx +a0y = g(x). If g(x) = 0, then the equation is called homogeneous.

  • 2. To solve a homogeneous Cauchy-Euler equation we set

y = xr and solve for r.

  • 3. The idea is similar to that for homogeneous linear

differential equations with constant coefficients.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-7
SLIDE 7

logo1 Overview An Example Double Check Discussion

Definition and Solution Method

  • 1. A second order Cauchy-Euler equation is of the form

a2x2d2y dx2 +a1xdy dx +a0y = g(x). If g(x) = 0, then the equation is called homogeneous.

  • 2. To solve a homogeneous Cauchy-Euler equation we set

y = xr and solve for r.

  • 3. The idea is similar to that for homogeneous linear

differential equations with constant coefficients. We will use this similarity in the final discussion.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-8
SLIDE 8

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-9
SLIDE 9

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-10
SLIDE 10

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-11
SLIDE 11

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-12
SLIDE 12

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-13
SLIDE 13

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-14
SLIDE 14

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-15
SLIDE 15

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-16
SLIDE 16

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-17
SLIDE 17

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-18
SLIDE 18

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-19
SLIDE 19

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 = 2r2 −r −1 =

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-20
SLIDE 20

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 = 2r2 −r −1 = r1,2 = −(−1)±

  • (−1)2 −4·2·(−1)

2·2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-21
SLIDE 21

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 = 2r2 −r −1 = r1,2 = −(−1)±

  • (−1)2 −4·2·(−1)

2·2 = 1±3 4

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-22
SLIDE 22

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 = 2r2 −r −1 = r1,2 = −(−1)±

  • (−1)2 −4·2·(−1)

2·2 = 1±3 4 = 1,−1 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-23
SLIDE 23

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 = 2r2 −r −1 = r1,2 = −(−1)±

  • (−1)2 −4·2·(−1)

2·2 = 1±3 4 = 1,−1 2 y(x) = c1x1 +c2x− 1

2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-24
SLIDE 24

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

2x2y′′ +xy′ −y = 2x2r(r −1)xr−2 +xrxr−1 −xr = 2r(r −1)xr +rxr −xr = 2r(r −1)+r −1 = 2r2 −r −1 = r1,2 = −(−1)±

  • (−1)2 −4·2·(−1)

2·2 = 1±3 4 = 1,−1 2 y(x) = c1x1 +c2x− 1

2 = c1x+c2

1 √x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-25
SLIDE 25

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-26
SLIDE 26

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-27
SLIDE 27

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-28
SLIDE 28

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-29
SLIDE 29

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2 2 = y′(1) = c1 − 1 2c2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-30
SLIDE 30

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2 2 = y′(1) = c1 − 1 2c2 −1 = 3 2c2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-31
SLIDE 31

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2 2 = y′(1) = c1 − 1 2c2 −1 = 3 2c2, c2 = −2 3

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-32
SLIDE 32

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2 2 = y′(1) = c1 − 1 2c2 −1 = 3 2c2, c2 = −2 3, c1 = 1−c2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-33
SLIDE 33

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2 2 = y′(1) = c1 − 1 2c2 −1 = 3 2c2, c2 = −2 3, c1 = 1−c2 = 5 3

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-34
SLIDE 34

logo1 Overview An Example Double Check Discussion

Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

y(x) = c1x+c2 1 √x y′(x) = c1 − 1 2c2 1 x

3 2

1 = y(1) = c1 +c2 2 = y′(1) = c1 − 1 2c2 −1 = 3 2c2, c2 = −2 3, c1 = 1−c2 = 5 3 y(x) = 5 3x− 2 3 1 √x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-35
SLIDE 35

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-36
SLIDE 36

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-37
SLIDE 37

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-38
SLIDE 38

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-39
SLIDE 39

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-40
SLIDE 40

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-41
SLIDE 41

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-42
SLIDE 42

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-43
SLIDE 43

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-44
SLIDE 44

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-45
SLIDE 45

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-46
SLIDE 46

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-47
SLIDE 47

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-48
SLIDE 48

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-49
SLIDE 49

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-50
SLIDE 50

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-51
SLIDE 51

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-52
SLIDE 52

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-53
SLIDE 53

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-54
SLIDE 54

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • + 1

x

1 2 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-55
SLIDE 55

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • + 1

x

1 2

  • (−1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-56
SLIDE 56

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • + 1

x

1 2

  • (−1)+ 1

3

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-57
SLIDE 57

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • + 1

x

1 2

  • (−1)+ 1

3 + 2 3

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-58
SLIDE 58

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • + 1

x

1 2

  • (−1)+ 1

3 + 2 3

  • ?

=

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-59
SLIDE 59

logo1 Overview An Example Double Check Discussion

Does y(x) = 5 3x− 2 3 1 √x Really Solve the Initial Value Problem 2x2y′′ +xy′ −y = 0, y(1) = 1, y′(1) = 2 y(x) = 5 3x− 2 3 1 √x , y(1) = 1 √ y′(x) = 5 3 + 1 3 1 x

3 2

, y′(1) = 2 √ y′′(x) = −1 2 1 x

5 2

2x2

  • −1

2 1 x

5 2

  • +x

5 3 + 1 3 1 x

3 2

  • −

5 3x− 2 3 1 √x

  • ?

= − 1 x

1 2

+ 5 3x+ 1 3 1 x

1 2

− 5 3x+ 2 3 1 x

1 2

?

= x 5 3 − 5 3

  • + 1

x

1 2

  • (−1)+ 1

3 + 2 3

  • ?

= √

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-60
SLIDE 60

logo1 Overview An Example Double Check Discussion

General Solution Method

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-61
SLIDE 61

logo1 Overview An Example Double Check Discussion

General Solution Method

  • 1. Inhomogeneous Cauchy-Euler equations are solved with

Variation of Parameters.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-62
SLIDE 62

logo1 Overview An Example Double Check Discussion

General Solution Method

  • 1. Inhomogeneous Cauchy-Euler equations are solved with

Variation of Parameters.

  • 2. A Cauchy-Euler equation is of the form

anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = g(x).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-63
SLIDE 63

logo1 Overview An Example Double Check Discussion

General Solution Method

  • 1. Inhomogeneous Cauchy-Euler equations are solved with

Variation of Parameters.

  • 2. A Cauchy-Euler equation is of the form

anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = g(x). If g(x) = 0, then the equation is called homogeneous.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-64
SLIDE 64

logo1 Overview An Example Double Check Discussion

General Solution Method

  • 1. Inhomogeneous Cauchy-Euler equations are solved with

Variation of Parameters.

  • 2. A Cauchy-Euler equation is of the form

anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = g(x). If g(x) = 0, then the equation is called homogeneous.

  • 3. The substitution t = ln(x) turns the Cauchy-Euler equation

anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = g(x) for x > 0 into a linear differential equation with constant coefficients.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-65
SLIDE 65

logo1 Overview An Example Double Check Discussion

General Solution Method

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-66
SLIDE 66

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-67
SLIDE 67

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-68
SLIDE 68

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.
  • 2. Cancel xr to obtain an equation p(r) = 0 for r.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-69
SLIDE 69

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.
  • 2. Cancel xr to obtain an equation p(r) = 0 for r.
  • 3. Find the solutions r1,...,rm of p(r) = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-70
SLIDE 70

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.
  • 2. Cancel xr to obtain an equation p(r) = 0 for r.
  • 3. Find the solutions r1,...,rm of p(r) = 0.
  • 4. If rk is real then y(x) = xrk solves the differential equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-71
SLIDE 71

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.
  • 2. Cancel xr to obtain an equation p(r) = 0 for r.
  • 3. Find the solutions r1,...,rm of p(r) = 0.
  • 4. If rk is real then y(x) = xrk solves the differential equation.
  • 5. If rk = ak +ibk is complex then y(x) = xak cos
  • bk ln(x)
  • and y(x) = xak sin
  • bk ln(x)
  • solve the differential equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-72
SLIDE 72

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.
  • 2. Cancel xr to obtain an equation p(r) = 0 for r.
  • 3. Find the solutions r1,...,rm of p(r) = 0.
  • 4. If rk is real then y(x) = xrk solves the differential equation.
  • 5. If rk = ak +ibk is complex then y(x) = xak cos
  • bk ln(x)
  • and y(x) = xak sin
  • bk ln(x)
  • solve the differential equation.
  • 6. If (r −rk)j is a factor of p(r), then xrk, ln(x)xrk, ...,
  • ln(x)

j−1xrk solve the differential equation. (If rk is complex we need to multiply the solutions from 5 with the power of the logarithm.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations

slide-73
SLIDE 73

logo1 Overview An Example Double Check Discussion

General Solution Method

Solving anxndny dxn +an−1xn−1dn−1y dxn−1 +···+a1xdy dx +a0y = 0.

  • 1. Substitute y = xr into the equation.
  • 2. Cancel xr to obtain an equation p(r) = 0 for r.
  • 3. Find the solutions r1,...,rm of p(r) = 0.
  • 4. If rk is real then y(x) = xrk solves the differential equation.
  • 5. If rk = ak +ibk is complex then y(x) = xak cos
  • bk ln(x)
  • and y(x) = xak sin
  • bk ln(x)
  • solve the differential equation.
  • 6. If (r −rk)j is a factor of p(r), then xrk, ln(x)xrk, ...,
  • ln(x)

j−1xrk solve the differential equation. (If rk is complex we need to multiply the solutions from 5 with the power of the logarithm.)

  • 7. The general solution is a linear combination of the

solutions above with generic coefficients c1,...,cn.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Cauchy-Euler Equations