COURSE SUMMARY

MATH2067: Differential Equations and Vector

Calculus for Engineers

Shane Leviton [Abstract]

1

Contents Ordinary Differential Equations .............................................................................................................. 7

Revision of first year (From MATH1903): ........................................................................................... 7

Exponential growth: ........................................................................................................................ 7

Logistic equation: ............................................................................................................................ 7

First order ODEs .............................................................................................................................. 7

First order linear differential equations: ......................................................................................... 8

Second order ODEs ......................................................................................................................... 9

First order systems: ....................................................................................................................... 12

Second Year ODEs: ............................................................................................................................ 13

First order linear differentiable equation: .................................................................................... 13

2nd order constant coefficient ODE’s ............................................................................................ 13

Non-Homogeneous case: .............................................................................................................. 14

Principle of superposition: ............................................................................................................ 14

Guessing functions: ....................................................................................................................... 14

Variation or Parameters: ................................................................................................................... 15

Functions needed to calculate ...................................................................................................... 15

Wronksian and fundamental solution .............................................................................................. 16

Notes on Wronksian function: ...................................................................................................... 16

Reduction of order: ........................................................................................................................... 16

Euler-Cauchy Equations: ................................................................................................................... 17

Most commonly: ........................................................................................................................... 17

Series Solutions of ODE’s: ................................................................................................................. 18

Techicality: .................................................................................................................................... 18

Notes: ............................................................................................................................................ 20

Method of Frobenius (regular singular points): ................................................................................ 21

Some definitions for Method of Frobenius: ................................................................................. 21

Notes on Frobenius compared to taylor series: ........................................................................... 22

Bessel’s Equation and Bessel Functions: ........................................................................................... 23

Bessel functions: ........................................................................................................................... 23

Radius of convergence: ..................................................................................................................... 24

Partial Differential Equations ................................................................................................................ 25

Boundary-Value Problems and Fourier Series Separation of varaibles for heat equation ............... 25

1 D heat equation: ........................................................................................................................ 25

Solving 1D heat equation: Separation of variables ....................................................................... 26

Fourier Series: (page 34 of notes) ..................................................................................................... 29

2

Complete set of orthogonality relations of sin/cos: ..................................................................... 29

Heat equation with heat flow out of 𝑥 = 𝐿: ..................................................................................... 31

Case 1: 𝑘 = 𝜆2: ............................................................................................................................. 31

Case 2: 𝑘 = 0 ................................................................................................................................ 31

Case 3: 𝑘 = −𝜆2: .......................................................................................................................... 31

Gibbs Phenomenon ........................................................................................................................... 32

Complex Fourier Series: .................................................................................................................... 34

Complex Fourier series for 𝑓(𝑥) ................................................................................................... 34

Sturm-Liouville Eigenvalue problems: .............................................................................................. 35

Regular Sturm Liouville Egienvalue Problems ............................................................................... 36

Sturm Liouville eigenvalue theorems: .......................................................................................... 36

Sturm Liouville theorems: ............................................................................................................. 36

Orthogonality of eigenvlaues: ....................................................................................................... 37

2D heat equation (heat conduction in a plate): ................................................................................ 39

Rectangular plate: ......................................................................................................................... 39

Circular plate: ................................................................................................................................ 39

Inhomogenoeous heat equation: ..................................................................................................... 41

Inhomogeneous boundary conditions: ......................................................................................... 41

+ heat source: ............................................................................................................................... 42

Inhomogeneous example 2:.......................................................................................................... 45

Application of heat equation: Daily and seasonal temperature variations in the earth .................. 46

transforms: ............................................................................................................................................ 48

Rookie example: ................................................................................................................................ 48

Finding inverse Laplace transforms .............................................................................................. 49

Overall picture to solve with Laplace Transforms: ........................................................................... 50

Suddenly heated half space: Solution by Laplace Transforms:......................................................... 50

Error function: ............................................................................................................................... 51

Properties of Laplace transforms: ..................................................................................................... 52

Shift theorem: ............................................................................................................................... 52

Convolution Theorem for Laplace Transforms: ............................................................................ 53

Laplace’s equation: ............................................................................................................................... 57

Comparison of Laplace’s equation to heat equation: ................................................................... 57

Laplaces (2D rectangular domain) .................................................................................................... 57

Laplaces for a circular disk (homogeneous): ..................................................................................... 60

Solving by separation of variables: ............................................................................................... 60

Revision of fluid flow: ....................................................................................................................... 62

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Conservation of mass (continuity equation):................................................................................ 62

Fourier Transform solution to heat equation: ...................................................................................... 64

Heat equation on infinite domain: .................................................................................................... 64

Fourier integral identity: ............................................................................................................... 66

Dirac delta function: ..................................................................................................................... 67

THE WAVE EQUATION ........................................................................................................................... 68

1D: ..................................................................................................................................................... 68

Solution: ........................................................................................................................................ 68

Vibrations in a non-uniform string .................................................................................................... 69

With function og 𝑐 ........................................................................................................................ 70

Spherical geometry and the wave equation: .................................................................................... 72

Let’s use separation of variables: 𝑢𝑟, 𝜃, 𝜙, 𝑡 = 𝑤𝑟, 𝜃, 𝜙ℎ𝑡 ........................................................... 73

Spherically symmetric waves in 3D ....................................................................................................... 77

VECTOR CALCULUS ................................................................................................................................ 78

Functions of many variables: ............................................................................................................ 78

Vector addition: ............................................................................................................................ 78

Scalar product: .............................................................................................................................. 78

Zero vector: ................................................................................................................................... 78

Basis vectors: ................................................................................................................................. 78

Scalar vector product .................................................................................................................... 78

Norm of vector: ................................................................................................................................. 79

Cauchy-Schwartz inequality: ............................................................................................................. 79

Angle between vectors: .................................................................................................................... 79

Projection vector: .............................................................................................................................. 79

Orthogonal vectors: .......................................................................................................................... 79

Area of parallelogram: .......................................................................................................................... 79

Jacobian matri: .................................................................................................................................. 79

Vector product (volume of parallelepiped) ...................................................................................... 80

Limits and continuity ............................................................................................................................. 80

Sequnces of vectors .......................................................................................................................... 80

Limit laws: ......................................................................................................................................... 80

Open and closed sets: ........................................................................................................................... 81

Interior point ..................................................................................................................................... 81

OPEN N-Ball: ...................................................................................................................................... 81

Open set: ........................................................................................................................................... 81

Functions of several variables ............................................................................................................... 81

4

Function limit rules: ...................................................................................................................... 81

Continuous functions: ....................................................................................................................... 82

Partial derivatives ................................................................................................................................. 82

Tangents to curves: ........................................................................................................................... 83

Tangent surfaces: .............................................................................................................................. 83

General product rules of differentiation: ............................................................................................. 83

Chain rule: ............................................................................................................................................. 83

Level sets : ............................................................................................................................................. 83

Gradient is perpendicular to level set: .................................................................................................. 84

Higher partial derivateies: .................................................................................................................... 84

Hessian matrix of 𝑓 at 𝒂: ....................................................................................................................... 84

Taylor polynomials: ............................................................................................................................... 84

Taylor’s theorem of the second order: ................................................................................................. 85

Jacobian matrix: .................................................................................................................................... 85

Differentiable functions ........................................................................................................................ 85

Maxima and minima ............................................................................................................................. 85

Global max min: ............................................................................................................................ 87

Mutltiple integrals ................................................................................................................................. 88

To calculate: ...................................................................................................................................... 88

Fubini’s therorem:................................................................................................................................. 88

Change of variables ............................................................................................................................... 88

Application: polar coordinates 𝑥, 𝑦 = 𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃 ............................................................................ 89

Triple integrals ...................................................................................................................................... 89

Transformation formula .................................................................................................................... 90

Application: spherical coordinates.................................................................................................... 90

Cylindrical coordinates:..................................................................................................................... 90

Line integrals ......................................................................................................................................... 91

Notation: ........................................................................................................................................... 91

Unit tangent: ..................................................................................................................................... 91

Arc length .......................................................................................................................................... 91

Line integrals of scalar functions ...................................................................................................... 92

Integrals of vector fields ................................................................................................................... 92

Vector fields ...................................................................................................................................... 93

Potential of 𝑓 .................................................................................................................................... 93

Closed vector field: ........................................................................................................................... 94

Curl .................................................................................................................................................... 94

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Curl in ℝ2: ......................................................................................................................................... 95

Path independence: .......................................................................................................................... 95

Theorem: ........................................................................................................................................... 95

Theorem ............................................................................................................................................ 97

Integral theorems in 2D ........................................................................................................................ 98

Domain and boundary: ..................................................................................................................... 98

Orientation: ....................................................................................................................................... 98

Green’s theorem ............................................................................................................................... 98

Application of green’s theorem: area of domain .......................................................................... 99

Application of green’s theorem: conservative vector fields ......................................................... 99

Stoke’s theorem in ℝ2 in the plane .................................................................................................. 99

circulation:’ ................................................................................................................................... 99

Flux .................................................................................................................................................... 99

Divergence of a function: .................................................................................................................... 100

Triple integrals .................................................................................................................................... 100

Fubini’s theorem for triple integrals: .................................................................................................. 100

Volume: ....................................................................................................................................... 100

Transformation formula in 3D: ....................................................................................................... 101

Eg cylindrical coordinates ............................................................................................................... 101

Eg 2 spherical coordinates: ............................................................................................................. 102

Surface integrals ................................................................................................................................. 103

Definition of surface ....................................................................................................................... 103

Orientation of surfaces and unit normal to surface ....................................................................... 103

6

Unit normal to implicity given surface: ........................................................................................... 104

Special vase of explicit surface: ...................................................................................................... 104

Unit normal to parametric surface ................................................................................................. 104

Calculation of surface integrals: ..................................................................................................... 104

Surface area of domain: .................................................................................................................. 105

Flux across surface: ............................................................................................................................. 105

Flux across graph: ........................................................................................................................... 106

Implicit representation of surfaces ................................................................................................. 106

Integral theorems in 3D: ..................................................................................................................... 106

Simple domain: ............................................................................................................................... 106

Divergence theorem: ...................................................................................................................... 106

Laplace operator: ................................................................................................................................ 107

Product rule: ................................................................................................................................... 107

Green’s first identity ........................................................................................................................... 107

Green’s second identity: ..................................................................................................................... 107

Stokes theorem for surfaces: .............................................................................................................. 107

Application: conservative vector fields in space ............................................................................. 108

7

Ordinary Differential Equations

Revision of first year (From MATH1903): ODE: has an unknown function of one variable and the derivatives of this function

Order= highest derivate

Exponential growth: 𝑑𝑁

𝑑𝑡= 𝑘𝑁

𝑁(𝑡) = 𝑁0𝑒𝑘𝑡

Logistic equation: 𝑑𝑁

𝑑𝑡= 𝑘𝑁(𝑡)(𝑀 − 𝑁(𝑡)) (𝑤𝑖𝑡ℎ max 𝑝𝑜𝑝 𝑀)

First order ODEs

Direction fields: (eg: logistic equation)

𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑎 𝑖𝑓𝑑𝑦

𝑑𝑥= 𝑓(𝑥) = 0

Separable differential equations:

Are in the form:

𝑑𝑦(𝑥)

𝑑𝑥= 𝑓(𝑥)

∴ 𝑦 = 𝐹(𝑥) + 𝐶 (𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛)

If given particular conditions: eg, (𝑥1, 𝑦1)

∫𝑦

𝑦1

𝑑𝑦 = ∫ 𝑓(𝑥)𝑥

𝑥1

𝑑𝑥

𝑦 = 𝐹(𝑥) + 𝑦1 − 𝐹(𝑥1)

𝑒𝑔:𝑑𝑦

𝑑𝑥= ln 𝑥 , 𝑎𝑛𝑑 (2,4)𝑖𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ

8

∫ 𝑑𝑦𝑦

4

Autonomous first order ode 𝑑𝑦

𝑑𝑥= 𝑔(𝑦)

∴𝑑𝑥

𝑑𝑦=

1

𝑔(𝑦) (𝑎𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑔(𝑦) ≠ 0)

𝑥 = ∫1

𝑔(𝑦)𝑑𝑦 (𝑎𝑛𝑑 𝑡ℎ𝑒𝑛 𝑔𝑒𝑡 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑦(𝑥) = 𝑓(𝑥) 𝑖𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒)

Separable ODE: 𝑑𝑦

𝑑𝑥= 𝑓(𝑥)𝑔(𝑦)

∫𝑑𝑦

𝑔(𝑦)= ∫ 𝑓(𝑥)𝑑𝑥

𝑡ℎ𝑒𝑛 𝐻(𝑦) = 𝑓(𝑥) + 𝐶, 𝑤ℎ𝑒𝑟𝑒 𝐻(𝑦) =1

𝐺(𝑦)

First order linear differential equations: 𝑑𝑦

𝑑𝑥= 𝑎(𝑥) + 𝑏(𝑥)𝑦

Change into: 𝑑𝑦

𝑑𝑥+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)

Multiply by an integrating factor: which has property that 𝑑𝐼

𝑑𝑥= 𝐼(𝑥)𝑃(𝑥)

: 𝐼(𝑥) = 𝑒∫ 𝑃(𝑥)𝑑𝑥

∴𝐼(𝑥)𝑑𝑦

𝑑𝑥+ 𝑃(𝑥)𝑦 𝐼(𝑥) = 𝑄(𝑥)𝐼(𝑥)

∴𝑑𝑦

𝑑𝑥𝐼(𝑥) + 𝑦

𝑑𝐼(𝑥)

𝑑𝑥= 𝑄(𝑥)𝐼(𝑥)

∴𝑑

𝑑𝑥(𝑦𝐼(𝑥)) = 𝑄(𝑥)𝐼(𝑥)

𝑦 =1

𝐼(𝑥)∫ 𝑄(𝑥)𝐼(𝑥)𝑑𝑥

𝑒𝑔:𝑑𝑦

𝑑𝑥+

2𝑦

𝑥=

1

𝑥𝑒𝑥2

𝐼(𝑥) = 𝑒∫ (

2𝑥

)𝑑𝑥= 𝑒2 ln 𝑥 = 𝑥2

∴𝑥2𝑑𝑦

𝑑𝑥+ 2𝑥𝑦 = 𝑥𝑒𝑥2

𝑑

𝑑𝑥(𝑥2𝑦) = 𝑥𝑒𝑥2

9

𝑥2𝑦 =1

2𝑒𝑥2

+ 𝐶

𝑦 =1

2𝑥2𝑒𝑥2

+𝐶

𝑥2

Classifications of ODEs:

- Separable

- Linear

- Separable and linear

- Neither

If neither separable and linear:

Multiply by a transformation variable:

𝑒𝑔 𝑦𝑑𝑦

𝑑𝑥= 𝑒−𝑥 − 𝑦2 (𝑛𝑒𝑖𝑡ℎ𝑒𝑟 𝑠𝑒𝑝𝑎𝑟𝑎𝑏𝑙𝑒 𝑜𝑟 𝑙𝑖𝑛𝑒𝑎𝑟)

𝑙𝑒𝑡 𝑧 = 𝑦2 ∴ 𝑑𝑧 = 2𝑦 𝑑𝑦

∴ 𝑦(

𝑑𝑧2𝑦)

𝑑𝑥= 𝑒−𝑥 − 𝑧

1

2

𝑑𝑧

𝑑𝑥= 𝑒−𝑥 − 𝑧

𝑑𝑧

𝑑𝑥+ 2𝑧 = 2𝑒−𝑥

𝑑

𝑑𝑥(𝑒2𝑥𝑧) = 2∫ 𝑒𝑥 𝑑𝑥

𝑒2𝑥𝑧 = 2𝑒𝑥 + 𝐶

𝑧 = 2𝑒−𝑥 + 𝐶𝑒−2𝑥

∴ 𝑦 = ±√𝑒−𝑥(2 + 𝐶𝑒−𝑥)

𝑜𝑡ℎ𝑒𝑟 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛𝑒𝑠: 𝑣 =𝑦

𝑥 𝑜𝑟 𝑤 = 𝑥 + 𝑦

Second order ODEs

Form 𝑎𝑦′′ + 𝑏𝑦′ + 𝑐𝑦 = 𝑑

Homogeneous:

𝑃(𝑥)𝑦′′ + 𝑄(𝑥)𝑦′ + 𝑅(𝑥)𝑦 = 0

2nd order linear homologous equations with constant coefficients

𝑎𝑦′′ + 𝑏𝑦′ + 𝑐𝑦 = 0

𝒚 = 𝒆𝒓𝒙

Eg:

10

𝑦′′ − 𝑦′ − 6𝑦 = 0

∴ 𝑎𝑢𝑥𝑖𝑙𝑙𝑎𝑟𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:

𝑟2 − 𝑟 − 6 = (𝑟 − 3)(𝑟 + 2) = 0 𝑟 = −3, 2

∴ 𝑦 = 𝐴𝑒−2𝑥 + 𝐵𝑒2𝑥

If auxiliary equation has 2 real distinct roots:

All fine

- If 2 complex roots:

- Eg: 𝑦′′ + 16𝑦 = 0

∴ 𝑟2 + 16 = 0 𝑟 = ±4𝑖

𝑦 = 𝐴𝑒4𝑖𝑥 + 𝐵𝑒−4𝑖𝑥

𝑏𝑢𝑡: 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑎𝑡 𝑒𝑖𝜃 = 𝑐𝑖𝑠𝜃 ∴

∴ 𝑦 = 𝐴𝑐𝑜𝑠(4𝑥) + 𝐵𝑠𝑖𝑛(4𝑥)

Eg 2: 𝑦′′ − 2𝑦′ + 5𝑦 = 0

∴ 𝑟2 − 2𝑟 + 5 = 0

𝑟 = 1 ± 2𝑖

∴ 𝑦(𝑥) = 𝑒𝑥(𝐴𝑐𝑜𝑠(2𝑥) + 𝐵𝑠𝑖𝑛(2𝑥))

1 root:

𝑒𝑔: 𝑦′′ − 6𝑦′ + 9𝑦 = 0

∴ 𝑟2 − 6𝑟 + 9 = 0 (𝑟 − 3)2 = 0 𝑟 = 3

∴ 𝑦(𝑥) = 𝐴𝑒3𝑥 + 𝐵𝑥𝑒3𝑥

2nd order linear non-homogenous differential equations with constant coefficients

Are: 𝑎𝑦′′ + 𝑏𝑦′ + 𝑐𝑦 = 𝐺(𝑥)

Step 1: find homologous solution of

𝑎𝑦′′ + 𝑏𝑦′ + 𝑐𝑦 = 0, 𝑢𝑠𝑖𝑛𝑔 𝑦 = 𝑒𝑟𝑥 𝑚𝑒𝑡ℎ𝑜𝑑 𝑎𝑏𝑜𝑣𝑒

Step 2: (particular solution)

- If polynomial 𝐺(𝑥) = 𝑔0 + ⋯ + 𝑔𝑛𝑥𝑛: use a general polynomial of same degree

- If exponential G(x) = 𝑒𝛼𝑥: use an expoenential of form

𝐾𝑒𝛼𝑥 (𝑤ℎ𝑒𝑟𝑒 𝛼 𝑖𝑠 𝑠𝑎𝑚𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥 𝑖𝑛 𝑒𝑥𝑝𝑜𝑒𝑛𝑡𝑛𝑡𝑖𝑎𝑙), if alpha was the solution to a

homologous solution, use 𝐾𝑥𝛼𝑥

- If G(x) = cos 𝑜𝑟 sin 𝜔𝑡 is trigonometrix (sin or cos): use 𝑦 = 𝐴𝑠𝑖𝑛(𝜔𝑡) + 𝐵𝑠𝑖𝑛(𝜔𝑡)

11

POLYNOMIAL:

𝑦′′ + 2𝑦′ − 3𝑦 = 𝑥2

𝐻𝑂𝑀𝑂𝐿𝑂𝐺𝑂𝑈𝑆 𝑆𝑂𝐿𝑈𝑇𝐼𝑂𝑁: (𝑎𝑏𝑜𝑣𝑒 𝑚𝑒𝑡ℎ𝑜𝑑): 𝑦ℎ = 𝐴𝑒𝑥 + 𝐵𝑒−3𝑥

𝑃𝐴𝑅𝑇𝐼𝐶𝑈𝐿𝐴𝑅: 𝑦𝑝 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶

∴ 2𝑎 + 2(2𝐴𝑥 + 𝐵) − 3(𝐴𝑥2 + 𝐵𝑥 + 𝐶) = 𝑥2

∴ −3𝐴𝑥2 + (4𝐴 − 3𝐵)𝑥 + (2𝐴 + 2𝐵 − 3𝐶) = 𝑥2

𝐴 = −1

3; 𝐵 = −

4

9; 𝐶 = −

14

27

∴ 𝑦 = 𝑦ℎ + 𝑦𝑝

𝑦(𝑥) = −1

3𝑥2 −

4

9𝑥 −

14

27+ 𝐴𝑒𝑥 + 𝐵𝑒−3𝑥

EXPONENTIAL:

𝑦′′ + 4𝑦 = 𝑒3𝑡

∴ 𝑟2 + 4 = 0; 𝑟 = ±2𝑖 ∴ 𝑦ℎ = 𝐴𝑠𝑖𝑛(2𝑡) + 𝐵𝑐𝑜𝑠(2𝑡)

𝑦𝑝 = 𝐾𝑒3𝑡

∴ 𝑒3𝑡(9𝐾 + 4𝑘) = 𝑒3𝑡; 𝐾 =1

13

∴ 𝑦(𝑡) =1

13𝑒3𝑡 + 𝐴𝑐𝑜𝑠(2𝑡) + 𝐵𝑠𝑖𝑛(2𝑡)

EXPONENTIAL 2:

𝑦′′ − 5𝑦′ + 6𝑦 = 𝑒2𝑥

ℎ𝑜𝑚𝑜𝑙𝑜𝑔𝑜𝑢𝑠: 𝑟2 − 5𝑟 + 6 = (𝑟 − 3)(𝑟 − 2) = 0

𝑟 = 2; 3

∴ 𝑦ℎ = 𝐴𝑒2𝑥 + 𝐵𝑒3𝑥

𝑁𝑂𝑇𝐸: 𝑤𝑒 𝑐𝑎𝑛𝑛𝑜𝑡 𝑢𝑠𝑒 𝑦𝑝 = 𝐾𝑒2𝑥𝑛𝑜𝑤, 𝑎𝑠 𝑖𝑡 𝑖𝑠 𝑖𝑛𝑐𝑙𝑢𝑑𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 ℎ𝑜𝑚𝑜𝑙𝑜𝑔𝑜𝑢𝑠, 𝑤𝑒 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑢𝑠𝑒

𝑦𝑝 = 𝐾𝑥𝑒2𝑘

𝑦′𝑝 = 𝐾(𝑒2𝑥 + 2𝑥𝑒2𝑥); 𝑦𝑝′′ = 𝐾(𝑒2𝑥 + 2(𝑒2𝑥 + 2𝑥𝑒2𝑥)) = 𝐾(3𝑒2𝑥 + 4𝑥𝑒2𝑥)

∴ 𝐾(3𝑒2𝑥 + 4𝑥𝑒2𝑥) − 5𝐾(𝑒2𝑥 + 2𝑥𝑒2𝑥) + 6𝐾𝑥𝑒3𝑥 = 𝑒2𝑥

3𝐾 − 5𝐾 = 1; 𝐾 = −1

∴ 𝑦(𝑥) = 𝐴𝑒2𝑥 + 𝐵𝑒3𝑥 − 𝑥𝑒2𝑥

If there is only 1 root of homologous: use 𝑦𝑝 = 𝐾𝑥2𝑒𝛼𝑥

TRIGONOMETRIC:

𝑥′′(𝑡) + 9𝑥(𝑡) = cos(𝛼𝑡)

𝑥𝑝(𝑡) = 𝐴𝑐𝑜𝑠(𝛼𝑡) + 𝐵𝑠𝑖𝑛(𝛼𝑡)

∴ 𝑥′′ + 9𝑥 = −(𝐴𝛼2 cos(𝛼𝑡) + 𝐵𝛼2 sin(𝛼𝑡)) + 9(𝐴𝑐𝑜𝑠(𝛼𝑡) + 𝐵𝑠𝑖𝑛(𝛼𝑡)) − cos 𝛼𝑡

∴ 𝐴 =1

9 − 𝛼2; 𝐵 = 0

∴ 𝑥 = 𝐴𝑐𝑜𝑠(3𝑡) + 𝐵𝑠𝑖𝑛(3𝑡 +1

9 − 𝛼2cos(𝛼𝑡)

12

First order systems: 𝑑𝑥

𝑑𝑡= 𝑓(𝑥, 𝑦);

𝑑𝑦

𝑑𝑡= 𝑔(𝑥, 𝑦) (𝑒𝑔 𝑝𝑟𝑒𝑑𝑎𝑡𝑜𝑟 𝑝𝑟𝑒𝑦 𝑠𝑦𝑠𝑡𝑒𝑚)

With constant coefficients: 𝑑𝑥

𝑑𝑡= 𝑎𝑥 + 𝑏𝑦

𝑑𝑦

𝑑𝑡= 𝑐𝑥 + 𝑏𝑦

Step 1: differentiate 1 with respect to t

Use simultaneous equations to substitute in 𝑥′𝑜𝑟 𝑦′ and 𝑥 𝑜𝑟 𝑦

Integrate

differentiate 1st solution

substitute

Eg: 𝑥′(𝑡) = 3𝑥 + 𝑦; 𝑦′(𝑡) = 2𝑥 − 4𝑦

∴ 𝑥′′(𝑡) = 3𝑥′(𝑡) + 𝑦′(𝑡); = 3𝑥′ + 2𝑥 − 4𝑦

∴ 𝑥′′(𝑡) − 3𝑥′(𝑡) − 2𝑥(𝑡) = −4𝑦

∴ 𝑥′′(𝑡) − 3𝑥′ − 2𝑥 = −4(𝑥′ − 3𝑥)

𝑥′′(𝑡) + 𝑥′(𝑡) + 10𝑥 = 0

∴ 𝑟2 + 𝑟 + 10 = 0

𝑟 = 𝑒𝑐𝑡

∴ 𝑥 = 𝐴𝑒𝑟1𝑡 + 𝐵𝑒𝑟2𝑡

∴ 𝑥′ = 𝐴𝑟1𝑒𝑟1𝑡 + 𝐵𝑟2𝑒𝑟2𝑡 = 3𝐴𝑒𝑟1𝑡 + 3𝐵𝑒𝑟2𝑡 + 𝑦

𝑦 = 𝐴(𝑟1 − 3)𝑒𝑟1𝑡 + 𝐵(𝑟2 − 3)𝑒𝑟2𝑡

∴ [𝑥(𝑡)

𝑦(𝑡)] =

13

Second Year ODEs:

First order linear differentiable equation: 𝑑𝑦

𝑑𝑥+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)

Integrating factor:

𝐼(𝑥) = 𝑒∫ 𝑃(𝑥)𝑑𝑥:

Eg:

𝑑𝑦

𝑑𝑥+

2

𝑥𝑦 =

1

𝑥𝑒𝑥2

∴ 𝐼(𝑥) = 𝑒∫ (

2𝑥

)𝑑𝑥 = 𝑒2 ln 𝑥 = 𝑥2

∴ 𝑥2𝑑𝑦

𝑑𝑥+ 2𝑥𝑦 = 𝑥𝑒𝑥2

𝑑

𝑑𝑥(𝑥2𝑦) = 𝑥𝑒𝑥2

∴ 𝑥2𝑦 =1

2𝑒𝑥2

+ 𝐶

𝑦(𝑥) =1

𝑥2(

1

2𝑒𝑥2

+ 𝐶)

2nd order constant coefficient ODE’s 𝑑2𝑦

𝑑𝑥2+ 𝑎

𝑑𝑦

𝑑𝑥+ 𝑏𝑦 = 0 (ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠)

𝑡𝑟𝑦: 𝑦 = 𝐶𝑒𝜆𝑥

∴ 𝐶𝑒𝑥(𝜆2 + 𝑎𝜆 + 𝑏) = 0

∴ 𝜆 = 𝜆1, 𝜆2

General solution depends on lambda:

Real and distinct lambda:

𝜆1,2 ∈ ℝ (𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙)

𝑦 = 𝐴𝑒𝜆1𝑥 + 𝐵𝑒𝜆2𝑥

Real non-distinct lambda

𝑦 = (𝐴 + 𝐵𝑥)𝑒𝜆2𝑥

Complex lambda

𝜆1,2 = 𝛼 ± 𝑖𝛽

𝑦 = 𝑒𝛼𝑥(𝐴 cos(𝛽𝑥) + 𝐵 sin(𝛽𝑥))

(𝐴, 𝐵 ∈ ℂ)

14

Non-Homogeneous case:

“Method of undetermined coefficients”: (guesswork)

𝑑2𝑦

𝑑𝑥2+ 3

𝑑𝑦

𝑑𝑥+ 2𝑦 = 6𝑥2 + 8:

Homogeneous case:

𝜆1 = −1; 𝜆2 = −2:

𝑦ℎ = 𝐴𝑒−𝑥 + 𝐵𝑒−2𝑥

Particular case:

𝑡𝑟𝑦 𝑦 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶

∴ (2𝐴) + 3(2𝐴𝑥 + 𝐵) + +2(𝐴𝑥2 + 𝐵𝑋 + 𝐶) = 6𝑥2 + 8

∴ 𝐴 = 3; 𝐵 = −9; 𝐶 =29

2

General solution:

𝑦 = 𝐴𝑒−𝑥 + 𝐵𝑒−2𝑥 + 3𝑥2 − 9𝑥 +29

2

Principle of superposition:

Constant:

𝑑2𝑦

𝑑𝑥2+ 𝐴(𝑥)

𝑑𝑦

𝑑𝑥+ 𝐵(𝑥)𝑦 = 0

If 𝑦1, 𝑦2 𝑎𝑟𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

𝑦 = 𝐶1𝑦1 + 𝐶2𝑦2

Is also a solution

Function:

𝑑2𝑦

𝑑𝑥2+ 𝐴(𝑥)

𝑑𝑦

𝑑𝑥+ 𝐵(𝑥)𝑦 = 𝐶(𝑥)

If 𝑦1 satisfies homogeneous equation (from above), and 𝑦2 satisfies inhomogenous: then

Consider:

�̂� = 𝐶𝑦1 + 𝑦2

∴ (𝐶𝑑2𝑦1

𝑑𝑥2+

𝑑2𝑦2

𝑑𝑥2 ) + 𝐴(𝑥) (𝐶𝑑𝑦1

𝑑𝑥+

𝑑𝑦2

𝑑𝑥) + 𝐵(𝑥)(𝐶𝑦1 + 𝑦2) = 0

∴ 𝐶 (𝑑2𝑦1

𝑑𝑥2+

𝐴(𝑥)𝑑𝑦1

𝑑𝑥+ 𝐵(𝑥)𝑦1) +

𝑑2𝑦2

𝑑𝑥2+ 𝐴(𝑥)

𝑑𝑦2

𝑑𝑥+ 𝐵(𝑥)𝑦2 = 𝐶(𝑥)

Guessing functions: - If polynomial 𝐺(𝑥) = 𝑔0 + ⋯ + 𝑔𝑛𝑥𝑛: use a general polynomial of same degree

15

- If exponential 𝐺(𝑥) = 𝑒𝛼𝑥: use an expoenential of form

𝐾𝑒𝛼𝑥 (𝑤ℎ𝑒𝑟𝑒 𝛼 𝑖𝑠 𝑠𝑎𝑚𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥 𝑖𝑛 𝑒𝑥𝑝𝑜𝑒𝑛𝑡𝑛𝑡𝑖𝑎𝑙), if alpha was the solution to a

homologous solution, use 𝐾𝑥𝛼𝑥

- If 𝐺(𝑥) = 𝑐𝑜𝑠 𝑜𝑟 𝑠𝑖𝑛 𝜔𝑡 is trigonometrix (sin or cos): use 𝑦 = 𝐴𝑠𝑖𝑛(𝜔𝑡) + 𝐵𝑠𝑖𝑛(𝜔𝑡)

Variation or Parameters: - Takes the ‘guesswork’ out, but it takes a while.

For:

𝑑2𝑦

𝑑𝑥2+ 𝐴(𝑥)

𝑑𝑦

𝑑𝑥+ 𝐵(𝑥) 𝑦 = 𝑓(𝑥)

If 𝑦1 and 𝑦2 are satisfy the homogenous ODE:

Eg:

𝑦 = 𝐶1(𝑥)𝑦1(𝑥) + 𝐶2(𝑥)𝑦2(𝑥)

Deriving and simplifying by subbing into 𝑑2𝑦

𝑑𝑥2 + 𝐴(𝑥)𝑑𝑦

𝑑𝑥+ 𝐵(𝑥) 𝑦 = 𝑓(𝑥)

We get: 𝐶1′𝑦1

′ + 𝐶2′ 𝑦2

′ = 𝑓(𝑥)

Try:

𝐶1′𝑦1 + 𝐶2

′ 𝑦2 = 0

∴ 𝑤𝑒 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟: 𝐶1′𝑦1 + 𝐶2

′ 𝑦2 = 0

And

Functions needed to calculate

∴ 𝐶1′ = −

𝑓(𝑥)𝑦2

𝑦1𝑦2′ − 𝑦2𝑦1

′ 𝑎𝑛𝑑 𝐶2′ =

𝑓(𝑥)𝑦1

𝑦1𝑦2′ − 𝑦2𝑦1

′

𝑦1𝑦2′ − 𝑦2𝑦1

′ 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑡ℎ𝑒 𝑊𝑟𝑜𝑛𝑠𝑘𝑒𝑖𝑛: 𝑊(𝑥)

∴ 𝐶1′ = −

𝑓(𝑥)𝑦1

𝑊(𝑥) ; 𝐶2

′ =𝑓(𝑥)𝑦1

𝑊(𝑥)

Example:

Eg:

𝑑2𝑦

𝑑𝑥2+ 3

𝑑𝑦

𝑑𝑥+ 2𝑦 = 6𝑥2 + 8:

Homogenous: 𝜆2 + 3𝜆 + 2 = 0:

∴ 𝜆 = −1, −2

∴ 𝑦1 = 𝑒−𝑥; 𝑦2 = 𝑒−2𝑥

∴ 𝑊(𝑥) = 𝑦1𝑦2′ − 𝑦2𝑦1

′ = −𝑒−3𝑥

16

∴ 𝐶1′ =

(6𝑥2 + 8)𝑒−2𝑥

−𝑒−3𝑥; 𝐶2

′ = −(6𝑥2 + 8)𝑒−𝑥

−𝑒−3𝑥

…

∴ 𝑦 = 3𝑥2 − 9𝑥 +29

2+ 𝐴𝑒−𝑥 + 𝐵𝑒−2𝑥

Variation of parameters works for all:

𝑑(𝑛)𝑦

𝑑𝑥(𝑛)+ ∑ 𝐴𝑖(𝑥)

𝑛−1

𝑖=0

𝑑(𝑖)𝑦

𝑑𝑥(𝑖)= 𝑓(𝑥)

Wronksian and fundamental solution For any

𝑦′′ + 𝑝(𝑡)𝑦′ + 𝑞(𝑡)𝑦 = 𝑔(𝑡)

Then:

(𝑦1(𝑡0) 𝑦2(𝑡0)

𝑦1′ (𝑡0) 𝑦2

′ (𝑡0)) (

𝐶1

𝐶2) = (

𝑦0

𝑦0)

Which has solutions if: det ((𝑦1(𝑡0) 𝑦2(𝑡0)

𝑦1′ (𝑡0) 𝑦2

′ (𝑡0))) ≠ 0

Notes on Wronksian function: - If 𝑊(𝑥) ≠ 0, then 𝑦1 𝑎𝑛𝑑 𝑦2 are linearly independent

- If 𝑦1, 𝑦2 are linearly dependent, then 𝑊(𝑥) = 0

- BUT: this DOES NOT mean that if 𝑦1 𝑎𝑛𝑑 𝑦2 are linearly independent, then 𝑊(𝑥) ≠ 0

necessarily happens

Reduction of order: Looking to solve variable coefficient linear equations of the form:

𝑑2𝑦

𝑑𝑥2+ 𝑎(𝑥)

𝑑𝑦

𝑑𝑥+ 𝑏(𝑥)𝑦 = {

0 (ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠)

𝑓(𝑥) (𝑖𝑛ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠)

Now that we have variation of parameters, we can just look at the homogeneous case, and then

apply variation of parameters to find the inhomogeneous

EG: if the function 𝑢(𝑥) is a solution to 𝑑2𝑦

𝑑𝑥2 + 𝑎(𝑥)𝑑𝑦

𝑑𝑥+ 𝑏(𝑥)𝑦 = 0; try the function

𝑦 = 𝑢(𝑥)𝑣(𝑥)

∴ 𝑦′ = 𝑢𝑣′ + 𝑣′𝑢

𝑦′′ = 𝑢𝑣′′ + 2𝑢′𝑣′ + 𝑣𝑢′′

17

∴ (𝑢𝑣′′ + 2𝑢′𝑣′ + 𝑣𝑢′′) + 𝑎(𝑥)(𝑢𝑣′ + 𝑣′𝑢) + 𝑏(𝑥)(𝑢𝑣) = 0

𝑣(𝑢′′ + 𝑎𝑢′ + 𝑏𝑢) + 𝑢𝑣′′ + (2𝑢′ + 𝑎𝑢)𝑣′ = 0

𝑎𝑠 𝑢 𝑠𝑜𝑙𝑣𝑒𝑠 𝑡ℎ𝑒 𝑂𝐷𝐸: 𝑢′′ + 𝑎𝑢′ + 𝑏𝑢 = 0 ∴ 𝑡ℎ𝑖𝑠 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑠 𝑡𝑜:

∴𝑣′′

𝑣′= − (𝑎 +

2𝑢′

𝑢)

Then integrate: exponentiate ect

Eg: reduction of order

solve (1 + 𝑥)𝑦′′ + 𝑥𝑦′ − 𝑦 = 0; given 𝑦 = 𝑥 is a solution:

𝑇𝑟𝑦 𝑦 = 𝑥𝑣(𝑥)

∴ 𝑦′ = 𝑥𝑣′ + 𝑣

𝑦′′ = 𝑥𝑣′′ + 2𝑣′

∴ (1 + 𝑥)(𝑥𝑣′′ + 2𝑣′) + 𝑥(𝑥𝑣′ + 𝑣) − 𝑥𝑣 = 𝑣′′(𝑥 + 𝑥2) + 𝑣′(2 + 2𝑥 + 𝑥2) = 0

𝑣′′

𝑣′= − (

𝑥2 + 2𝑥 + 2

𝑥2 + 𝑥) = − (1 +

𝑥 + 2

𝑥2 + 𝑥) = (−1 −

2

𝑥+

1

1 + 𝑥)

∴ ln 𝑣′ = −𝑥 − 2 ln 𝑥 + ln(1 + 𝑥) + �̂�

= ln (𝐶̅(1 + 𝑥)𝑒−𝑥

𝑥2 )

𝑣′ =𝐶̅(1 + 𝑥)𝑒−𝑥

𝑥2

∴ 𝑣 = ∫ 𝐶̅𝑑

𝑑𝑥(

𝑒−𝑥

𝑥)

𝑣 =𝐶𝑒−𝑥

𝑥+ 𝐷

∴ 𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝑦 = 𝑥𝑣 = 𝐶𝑒−𝑥 + 𝐷𝑥

Euler-Cauchy Equations: Are of the form:

∑ 𝑎𝑝𝑥𝑝𝑑(𝑝)𝑦

𝑑𝑥(𝑝)

𝑛

𝑝=0

= {0

𝑓(𝑥)

Most commonly:

𝑥2𝑑2𝑦

𝑑𝑥2+ 𝑎𝑥

𝑑𝑦

𝑑𝑥+ 𝑏𝑦 = 0

(can solve = 𝑓(𝑥) with variation of parameters)

Try:

𝑦 = 𝑘𝑥𝜆

∴ 𝑥2(𝑘𝜆(𝜆 − 1)𝑥𝜆−2) + 𝑎𝑥(𝜆𝑘𝑥𝜆−1) + 𝑏(𝑘𝑥𝜆) = 0

𝐴𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑙 ≠ 0; 𝑥 ≠ 0 (𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛)

𝜆(𝜆 − 1) + 𝑎𝜆 + 𝑏 = 𝜆2 + (𝑎 − 1)𝜆 + 𝑏 = 0

18

∴ 𝜆1,2 =(1 − 𝑎 ± √(𝑎 − 1)2 − 4𝑏)

2

Real, distinct lambda:

If 𝜆1, 𝜆2 𝑑𝑖𝑠𝑡𝑖𝑛𝑐𝑡 𝑖𝑛 ℝ:

𝑦 = 𝐴𝑥𝜆1 + 𝐵𝑥𝜆2

Distinct, complex lambda:

𝜆1 ≠ 𝜆2 ∈ ℝ

∴ 𝜆 = 𝛼 ± 𝑖𝛽

∴ 𝑦 = 𝐴𝑥𝛼+𝑖𝛽 + 𝐵𝑥𝛼−𝑖𝛽 = 𝑥𝛼 (𝐴(𝑒ln 𝑥)𝑖𝛽

+ 𝐵(𝑒ln 𝑥)−𝑖𝛽

)

𝑦 = 𝑥𝛼(𝐴 cos(ln 𝛽𝑥) + 𝐵 sin(ln 𝛽𝑥))

(where 𝐴, 𝐵 ∈ ℂ)

Equal roots:

If 𝜆1 = 𝜆2 = 𝜆

𝑦 = 𝐴𝑥𝜆 + 𝐵𝑥𝜆 ln 𝑥

Example of E-C equation:

Solve 𝑥2 𝑑2𝑦

𝑑𝑥2 − 5𝑥𝑑𝑦

𝑑𝑥+ 10 𝑦 = 0

𝑇𝑟𝑦 𝑦 = 𝑘𝑥𝜆

∴ 𝜆(𝜆 − 1) − 5𝜆 + 10 = 𝜆2 − 6𝜆 + 10 = 0

∴ 𝜆 =6 ± √36 − 40

2= 3 ± 𝑖

∴ 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠: 𝑦 = 𝑥3(𝐴 cos(ln 𝑥) + 𝐵 sin(ln 𝑥))

Series Solutions of ODE’s: Uses the Tayloer series:

𝑦 = ∑ 𝑐𝑛𝑥𝑛

∞

𝑛=0

(𝑇𝑎𝑦𝑙𝑜𝑟 𝑆𝑒𝑟𝑖𝑒𝑠 𝑎𝑏𝑜𝑢𝑡 𝑜𝑟𝑖𝑔𝑖𝑛, 𝑠𝑜𝑚𝑒𝑡𝑖𝑚𝑒𝑠 𝑠𝑜𝑚𝑒𝑤ℎ𝑒𝑟𝑒 𝑒𝑙𝑠𝑒)

Techicality: If the series converges uniformly, the series can be differentiated term by term. (This will be

assumed in this course).

Eg:

19

Find the general solution of:

𝑦′′ + 𝑥𝑦′ + 𝑦 = 0:

Let:

𝑦 = ∑ 𝑐𝑛𝑥𝑛

∞

𝑛=0

𝑦′ = ∑ 𝑛 𝑐𝑛𝑥𝑛−1

∞

𝑛=0

= ∑ 𝑛 𝑐𝑛𝑥𝑛−1

∞

𝑛=1

𝑦′′ = ∑ 𝑛(𝑛 − 1)𝑐𝑛𝑥𝑛−2

∞

𝑛=0

= ∑ 𝑛(𝑛 − 1)𝑐𝑛𝑥𝑛−2

∞

𝑛=2

∴ ∑ 𝑛(𝑛 − 1)𝑐𝑛𝑥𝑛−2

∞

𝑛=2

+ 𝑥 ∑ 𝑛 𝑐𝑛𝑥𝑛−1

∞

𝑛=1

+ ∑ 𝑐𝑛𝑥𝑛

∞

𝑛=0

= 0

Shifting indicies, so we can equate powers of 𝑥:

∴ ∑(𝑛 + 2)(𝑛 + 1)𝑐𝑛+2𝑥𝑛

∞

𝑛=0

+ ∑ 𝑛 𝑐𝑛𝑥𝑛−1

∞

𝑛=1

+ ∑ 𝑐𝑛𝑥𝑛

∞

𝑛=0

= 0

→ 2𝑐2 + 𝑐0 + ∑((𝑛 + 2)(𝑛 + 1)𝑐𝑛+2 + (𝑛 + 1)𝑐𝑛)𝑥𝑛

∞

𝑛=1

= 0

(𝑡ℎ𝑒 2𝑐2 𝑎𝑛𝑑 𝑐0 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑢𝑚′𝑠 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑎𝑡 𝑛 = 0)

∴ 𝑎𝑠 𝑒𝑎𝑐ℎ 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑥 𝑚𝑢𝑠𝑡 = 0:

2𝑐2 + 𝑐0 = 0 ⟹ 𝑐2 = −𝑐0

2

𝑎𝑛𝑑 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑥𝑛 = 0:

𝑐𝑛+2 = −(𝑛 + 1)𝑐𝑛

(𝑛 + 2)(𝑛 + 1)= −

𝑐𝑛

𝑛 + 2

∴ 𝑐0 𝑎𝑛𝑑 𝑐1 𝑎𝑟𝑒 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦; 𝑠𝑜 𝑜𝑑𝑑 𝑑𝑒𝑝𝑒𝑛𝑑 𝑜𝑛 𝑐1, 𝑒𝑣𝑒𝑛 𝑜𝑛 𝑐0

∴ 𝑦 = ∑ 𝑐𝑛𝑥𝑛

∞

𝑛=0

= 𝑐0 + 𝑐1𝑥 + 𝑐2𝑥2 + ⋯

𝑐2 = −𝑐0

2; 𝑐4 = −

𝑐2

4=

𝑐0

8…

𝑐3 = −𝑐1

3; 𝑐5 = −

𝑐3

5=

𝑐1

15…

𝑦(0) = 𝑐0; 𝑦′(0) = 𝑐1

Hence:

𝑦 = 𝑐0 (1 −𝑥2

2+

1

8𝑥4 − ⋯ ) + 𝑐1 (𝑥 −

1

3𝑥3 + ⋯ )

20

= 𝑦(0) (1 −𝑥2

2+

1

8𝑥4 − ⋯ ) + 𝑦′(0) (𝑥 −

1

3𝑥3 + ⋯ )

𝑛𝑜𝑡𝑒: (1 −𝑥2

2+

1

8𝑥4 − ⋯ ) = 𝑒−

𝑥2

2

Now we use reduction of order to find what (𝑥 −1

3𝑥3 + ⋯ )

∴ 𝑙𝑒𝑡 𝑦 = 𝑒−𝑥2

2 𝑣(𝑥) → 𝑦′ = 𝑒−𝑥2

2 (𝑣′ − 𝑣𝑥)

∴ 𝑆𝐼𝑚𝑝𝑙𝑖𝑓𝑦𝑖𝑛𝑔 𝑎𝑛𝑑 𝑠𝑢𝑏𝑏𝑖𝑛𝑔 𝑎𝑛𝑑 𝑡ℎ𝑖𝑛𝑔𝑠:

𝑣′′ − 𝑥𝑣′ = 0

𝑣′′

𝑣′= 𝑥

ln 𝑣′ =𝑥2

2+ 𝐴1

𝑣′ = 𝐴2𝑒−𝑥2

2

𝑣 = 𝐴2 ∫ 𝑒−𝑥2

2 𝑑𝑥

∴ 2𝑛𝑑 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 (𝑥 −1

3𝑥3 + ⋯ ) = 𝑒−

𝑥2

2 ∫ 𝑒−𝑥2

2 𝑑𝑥

∴ 𝑦 = 𝐴𝑒−𝑥2

2 (𝑒−𝑥2

2 ∫ 𝑒−𝑥2

2 𝑑𝑥) = 𝐴𝑒−𝑥2∫ 𝑒−

𝑥2

2 𝑑𝑥

Notes: - You want the series to converge to truly consider it a representation of the solution.

(divergent in another course)

Ratio test for convergence:

𝑓𝑜𝑟 𝑦 = ∑ 𝑐𝑛𝑡𝑛

∞

𝑛=0

:

𝐿 = lim𝑛→∞

|𝑐𝑛+1𝑡𝑛+1

𝑐𝑛𝑡𝑛| = lim

𝑛→∞|𝑐𝑛+1

𝑐𝑛| |𝑡|

𝑖𝑓 𝐿 < 1: 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠

𝐿 > 1: 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠

𝐿 = 1: 𝑛𝑜𝑡 𝑒𝑛𝑜𝑢𝑔ℎ 𝑖𝑛𝑓𝑜

21

This technique will work for any 𝑎(𝑥)𝑎𝑛𝑑 𝑏(𝑥) themselves admit converging Taylor series

expansions at 𝑥 = 0; convergent with some radius |𝑥| < 𝑅

Method of Frobenius (regular singular points):

Some definitions for Method of Frobenius: If 𝑦′′ + 𝑃(𝑥)𝑦′ + 𝑄(𝑥)𝑦 = 0:

- If 𝑃(𝑥)𝑎𝑛𝑑 𝑄(𝑥) remain finite at 𝑥 = 𝑥0, 𝑥0 is called an ordinary point

- If either 𝑃(𝑥) or 𝑄(𝑋) divereges as 𝑥 → 𝑥0, then 𝑥0 is called a singular point

- If either 𝑃(𝑥) or 𝑄(𝑥) diverges as 𝑥 → 𝑥0, but (𝑥 − 𝑥0)𝑃(𝑥) and (𝑥 − 𝑥0)2𝑄(𝑥) remains

finite as 𝑥 → 𝑥0, 𝑥 = 𝑥0 is a regular singular point

Consider (note: singularity at 𝑥 = 0) :

𝑦′′ +𝛼(𝑥)

𝑥𝑦′ +

𝛽(𝑥)

𝑥2𝑦 = 0

∴ 𝑡𝑖𝑚𝑒𝑠 𝑏𝑦 𝑥2

(𝑥 = 0 is a regular singular point)

NOW: seek solutions of the form:

𝑦 = 𝑥𝑟 ∑ 𝑐𝑛𝑥𝑛

∞

𝑛=0

= ∑ 𝑐𝑛𝑥𝑛+𝑟

∞

𝑛=0

𝑒𝑥𝑎𝑚𝑝𝑙𝑒:

2𝑥2𝑦′′ − 𝑥𝑦′ + (1 + 𝑥)𝑦 = 0

𝑐0 ≠ 0; 𝑥 > 0

→ 𝑦′ = ∑(𝑛 + 𝑟)𝑐𝑛𝑥𝑛+𝑟−1

∞

𝑛=0

; 𝑦′′ = ∑(𝑛 + 𝑟)(𝑛 + 𝑟 − 1)𝑐𝑛𝑥𝑛+𝑟−2

∞

𝑛=0

∴ 2 ∑ 𝑐𝑛(𝑛 + 𝑟)(𝑛 + 𝑟 − 1)𝑥𝑛+𝑟

∞

𝑛=0

− ∑ 𝑐𝑛(𝑛 + 𝑟)𝑥𝑛+𝑟

∞

𝑛=0

+ (1 + 𝑥) ∑ 𝑐𝑛𝑥𝑛+𝑟

∞

𝑛=0

= 0

∴ ∑ 𝑐𝑛[(2𝑛 + 2𝑟 − 1)(𝑛 + 𝑟 − 1)]𝑥𝑛+𝑟

∞

𝑛=0

+ ∑ 𝑐𝑛𝑥𝑛+𝑟+1

∞

𝑛=0

= 0

∴ 𝑥𝑟 {(2𝑟 − 1)(𝑟 − 1)𝑐0 + ∑(𝑐𝑛(2𝑛 + 2𝑟 − 1)(𝑛 + 𝑟 − 1) + 𝑐𝑛−1)𝑥𝑛)

∞

𝑛=1

} = 0

Radius of convergence

The lowest power is 𝑥𝑟, which gives a solution of 𝑟 =1

2, 1

22

∴ 𝑐𝑛(2𝑛 + 2𝑟 − 1)(𝑛 + 𝑟 − 1) + 𝑐𝑛−1 = 0

∴ 𝑐𝑛 = −𝑐𝑛−1

(𝑛 + 𝑟 − 1)(2𝑛 + 2𝑟 − 1)

For 𝑟 = 1:

𝑐𝑛 = −𝑐𝑛−1

𝑛(2𝑛 + 1)

∴ 𝑦1 = 𝑐0𝑥(1 −1

1 × 3𝑥 +

1

2 × 3 × 5𝑥3 − (

1

1 × 2 × 3 × 5 × 7) 𝑥3 + ⋯ )

For 𝑟 =1

2:

𝑐𝑛 = −𝑐𝑛−1

𝑛(2𝑛 − 1)

∴ 𝑐1 = −𝑐0; 𝑐2 = −𝑐1

3 ∙ 2=

𝑐0

2 × 3;

∴ 𝑦2 = 𝑐0𝑥12 (1 − 𝑥 +

1

6𝑥2 + ⋯ )

∴ 𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 2𝑥2𝑦′′ − 𝑥𝑦′ + (1 + 𝑥)𝑦 = 0 𝑖𝑠:

𝑦 = 𝐴𝑥 (1 −1

1 ∙ 3𝑥 + ⋯ ) + 𝐵𝑥

12(1 − 𝑥 + ⋯ )

Radius of convergence:

For 𝑟 = 1: 𝑐𝑛 = −𝑐𝑛−1

𝑛(2𝑛+1)

∴ lim𝑛→∞

|𝑐𝑛𝑥𝑛

𝑐𝑛−1𝑥𝑛−1| = lim𝑛→∞

(|−𝑐𝑛−1

𝑛(2𝑛 + 1)𝑐𝑛−1| |𝑥|)

= lim𝑛→∞

(|−1

2𝑛2 + 𝑛| |𝑥|) → 0 < 1

∴ 𝑅 → ∞

For 𝑟 =1

2: 𝑐𝑛 = −

𝑐𝑛−1

𝑛(2𝑛−1)

∴ lim𝑛→∞

|𝑐𝑛𝑥𝑛

𝑐𝑛−1𝑥𝑛−1| = lim𝑛→∞

(|−𝑐𝑛−1

𝑛(2𝑛 − 1)𝑐𝑛−1| |𝑥|)

= lim𝑛→∞

(|−1

2𝑛2 − 1| |𝑥|) → 0 < 1

∴ 𝑅 → ∞

Notes on Frobenius compared to taylor series: Not: for frobenius, as 𝑦 = ∑ 𝑐𝑛𝑥𝑛+𝑟∞

𝑛=0 , the terms WILL NOT disappear when derived (unlike taylor

series), because of the extra 𝑥𝑟 term

- If 𝑟′𝑠 differ by an integer, use the higher value of 𝑟, and then use reduction of order: 𝑦 = 𝑢𝑣

23

Bessel’s Equation and Bessel Functions: Another famous equation that pops up everywhere.

Consider:

𝑦′′ +1

𝑥𝑦′ + (1 −

𝑝2

𝑥2) 𝑦 = 0

𝑝 is the ORDER of the Bessel equation. It doesn’t have to be an integer, but it usually is.

𝑥 = 0 is a regular singular point and so the equation is a candidate for a frobenius series expansion

Try:

𝑦 = ∑ 𝑐𝑛𝑥𝑛+𝑟

∞

𝑛=0

𝑠𝑢𝑏 𝑖𝑛𝑡𝑜 𝑥2𝑦′′ + 𝑥𝑦′ + (𝑥2 − 𝑝2)𝑦 = 0

∴ ∑ 𝑐𝑛(𝑛 + 𝑟)(𝑛 + 𝑟 − 1)𝑥𝑛+𝑟

∞

𝑛=0

+ ∑ 𝑐𝑛𝑥𝑛+𝑟+2

∞

𝑛=0

+ ∑ 𝑐𝑛(𝑛 + 𝑟)𝑥𝑛+𝑟

∞

𝑛=0

− 𝑝2 ∑ 𝑐𝑛𝑥𝑛+𝑟

∞

𝑛=0

= 0

∴ ∑ 𝑐𝑛(𝑛 + 𝑟)(𝑛 + 𝑟 − 1)𝑥𝑛+𝑟

∞

𝑛=0

+ ∑ 𝑐𝑛−2𝑥𝑛+𝑟

∞

𝑛=2

− 𝑝2 ∑ 𝑐𝑛𝑥𝑛+𝑟

∞

𝑛=0

= 0

→ : 𝑐0(𝑟2 − 𝑟 + 𝑟 − 𝑝2)𝑥𝑟 + 𝑐1(𝑟2 + 𝑟 + 𝑟 + 1 − 𝑝2)𝑥𝑟+1

+ ∑(𝑐𝑛(𝑛 + 𝑟)(𝑛 + 𝑟 − 1) + 𝑐𝑛(𝑛 + 𝑟) + 𝑐𝑛−2 − 𝑝2𝑐𝑛)𝑥𝑛+𝑟

∞

𝑛=2

= 0

𝑐0: ∴ 𝑟 = ±𝑝

𝑐1: 𝑟 = −1

2

𝑥: [(𝑛 + 𝑟)(𝑛 + 𝑟 − 1 + 1) − 𝑝2]𝑐𝑛 = 𝑐𝑛−2

𝑐𝑛 = −𝑐𝑛−2

(𝑛 + 𝑟)2 − 𝑝2

∴ 𝑟 = 𝑝: 𝑐𝑛 = −𝑐𝑛−2

𝑛(𝑛 + 2𝑝)

𝑡ℎ𝑖𝑠 𝑠𝑜𝑙𝑣𝑒𝑠 𝑡𝑜 𝑏𝑒:

Bessel functions:

𝑦 = 𝑥𝑝𝐶0 (1 + ∑(−1)𝑚𝑥2𝑚

𝑚! 22𝑚(1 + 𝑝)(2 + 𝑝) … (𝑚 + 𝑝)

∞

𝑚=1

)

24

Radius of convergence:

lim𝑛→∞

|𝑐𝑛𝑥𝑛

𝑐𝑛−2𝑥𝑛−2| = lim𝑛→∞

|𝑐𝑛−2

𝑐𝑛−2((𝑛 − 𝑝)2 − 𝑝2)| |𝑥2|

∴ 𝑅 → ∞

Normalisation of Bessel function (with gamma function)

Γ(𝑝) = ∫ 𝑡𝑝−1𝑒−𝑡𝑑𝑡∞

0

, 𝑝 > 0 (𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑠𝑎𝑡𝑖𝑜𝑛)

→ Γ(𝑝) = (𝑝 − 1)Γ(𝑝 − 1)

Γ (1

2) = √𝜋 = (−

1

2) !

∴ 𝐶0 =1

2𝑝Γ(𝑝 + 1) (𝑔𝑜𝑖𝑛𝑔 𝑏𝑎𝑐𝑘 𝑡𝑜𝑒 𝐵𝑒𝑠𝑠𝑒𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛)

∴ 𝑦 = 𝑥𝑝1

2𝑝Γ(𝑝 + 1)(1 + ∑

(−1)𝑚𝑥2𝑚

𝑚! 22𝑚(1 + 𝑝)(2 + 𝑝) … (𝑚 + 𝑝)

∞

𝑚=1

)

= 𝑥𝑝(1

2𝑝Γ(𝑝 + 1)(1 + ∑

(−1)𝑚𝑥2𝑚

𝑚! Γ(𝑝 + 𝑚 + 1)

∞

𝑚=1

)

𝐵𝑒𝑠𝑠𝑒𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 𝑝 = (𝑥

2)

𝑝

(∑(−1)𝑛 (

𝑥2)

2𝑛

𝑛! Γ(𝑝 + 𝑛 + 1)

∞

𝑛=0

) = 𝐽𝑝(𝑥)

Replacing 𝑝 by −𝑝 (as 𝑟 = ±𝑝) , yields the 2nd solution:

𝐽−𝑝(𝑥) = (2

𝑥)

𝑝

(∑(−1)𝑛 (

𝑥2)

2𝑛

𝑛! Γ(−𝑝 + 𝑛 + 1)

∞

𝑛=0

)

But:

Γ(𝑝) = (𝑝 − 1)Γ(𝑝 − 1)

∴ Γ(1) = 0Γ(0) = 0! ∴→ Γ(0) = ∞ → (−𝑛)! = ∞ (𝑓𝑜𝑟 𝑛 ∈ ℕ exluding {0})

So: if 𝑝 is an integer:

𝐽−𝑝(𝑥) = (2

𝑥)

𝑝

(∑(−1)𝑛 (

𝑥2)

2𝑛

𝑛! Γ(−𝑝 + 𝑛 + 1)

∞

𝑛=𝑝

)

25

Let 𝑚 = 𝑛 − 𝑝

= (2

𝑥)

𝑝

∑(−1)𝑚+𝑝 (

𝑥2

)2𝑛+2𝑝

(𝑚 + 𝑝)! Γ(𝑚 + 1)

∞

𝑚=0

= (−1)𝑝 (𝑥

2)

2𝑝

(2

𝑥)

𝑝

∑(−1)𝑚 (

𝑥2

)2𝑚

𝑚! Γ(𝑚 + 𝑝 + 1)

∞

𝑚=0

= (−1)𝑝 𝐽𝑝(𝑥)

Therefore if 𝑝 is an integer: then the 2 solutions ARE NOT linearly independent independent, so we’d

only have 1 solution.

In which case:

𝑌𝑝(𝑥) =cos(𝜋𝑝) 𝐽𝑝(𝑥) − 𝐽−𝑝(𝑥)

sin(𝜋𝑝)

Partial Differential Equations

Boundary-Value Problems and Fourier Series Separation of varaibles for heat equation - Consider the problem of heat conduction in a 1 dimensional bar. With a given initial

temperature distribution and temperature held constant at each end.

- Let 𝑢(𝑥, 𝑡) be the temperature difference between the actual temperature and the constant

at the end, then for all 𝑡 ≥ 0:

1 D heat equation: 𝜕𝑢

𝜕𝑡= 𝜅

𝜕2𝑢

𝜕𝑥2

(𝜅 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦)

Notes on heat equation:

In higher dimensions, the equation is:

𝜕𝑢

𝜕𝑡= 𝜅 ∇2𝑢

Have “correct” amount of Initial conditions and boundary conditions

𝑥 = 0; 𝑢 = 0

𝑥 = 0; 𝑢 = 0 𝑥 = 𝐿; 𝑢 = 0

26

Solving 1D heat equation: Separation of variables Separations of variables usually requires linear and homogeneous PDE with linear and homogeneous

Boundary conditions

Using separation of variables:

Let

𝑢 = 𝑋(𝑥)𝑇(𝑡)

∴𝜕𝑢

𝜕𝑡= 𝜅

𝜕2𝑢

𝜕𝑥2

⟹ 𝑋𝑇′ = 𝜅𝑋𝑇′′

∴𝑇′

𝜅𝑇=

𝑋′′

𝑋= 𝑘

(𝑠𝑜𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑎𝑠 𝑡ℎ𝑒 𝑜𝑛𝑙𝑦 𝑤𝑎𝑦 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡 𝑐𝑎𝑛 𝑒𝑞𝑢𝑎𝑙 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑥 𝑖𝑠 𝑖𝑓 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)

𝑘 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑖𝑡ℎ𝑒𝑟 + ,0 𝑜𝑟 −

Positive k: (trivial soluiton)

If 𝑘 is positive

𝑘 = 𝜆2 (𝜆 > 0)

∴ 𝑋′′ = 𝜆2𝑋

𝑋 = 𝐴𝑒𝜆𝑥 + 𝐵𝑒−𝜆𝑥

Boundary conditions:

𝑢 = 0, 𝑎𝑡 𝑥 = 0, 𝐿

𝑥 = 0: 𝑢(0, 𝑡) = 𝑋(0)𝑇(𝑡) = 0 → 𝑋(0) = 0

𝑥 = 𝑙: 𝑢(𝐿, 𝑡) = 𝑋(𝐿)𝑇(𝑡) = 0 → 𝑋(𝐿) = 0

∴ 𝑋(0) = 0; → 𝐴 + 𝐵 = 0;

𝑋(𝐿) = 0: 𝐴𝑒𝜆𝐿 − 𝐴𝑒−𝜆𝐿 = 0

∴ 𝐴 = 𝐵 = 0

Trivial solution.

∴ boring 0 solution

𝑘 = 0 trivial Solution:

𝑋′′ = 0

→ 𝑋 = 𝐴𝑥 + 𝐵

𝑋(0) = 0 → 𝐵 = 0

𝑋(𝐿) = 0 → 𝐴 = 0

27

𝑘 < 0 Solution:

𝑘 = −𝜆2 (𝑓𝑜𝑟 𝑠𝑜𝑚𝑒𝜆 > 0)

∴ 𝑋′′ + 𝜆2𝑋 = 0 𝑋 = 𝐴 cos(𝜆𝑥) + 𝐵 sin(𝜆𝑥)

𝑋(0) = 0 → 𝐴 = 0

𝑋(𝐿) = 0 → 𝐵 = 0 𝑜𝑟 sin(𝜆𝐿) = 0

∴ 𝜆 =𝑛𝜋

𝐿 (𝑓𝑜𝑟 𝑛 = 1,2,3 … ) (𝑜𝑛𝑙𝑦 𝑡𝑎𝑘𝑒 𝑛 ∈ ℤ+ 𝑎𝑠 𝜆, 𝐿 > 0)

∴⟹𝑇′

𝜅𝑇= 𝑘 = −𝜆2 = −

𝑛2𝜋2

𝐿2

∴ 𝑇′ = −𝑛2𝜋2

𝐿2 𝜅𝑇

(i.e. exponential decay)

∴ 𝑇 = 𝐶𝑛 𝑒−

𝑛2𝜋2

𝐿2 𝜅𝑡

(𝐶𝑛 as constant term will be detirmed by your 𝑛 eigenvalue values)

𝑢 = 𝑋𝑇 = 𝐴𝑛 sin (𝑛𝜋

𝐿𝑥) 𝑒

−𝑛2𝜋2

𝐿2 𝜅𝑡 (𝑓𝑜𝑟 𝑎𝑛𝑦 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑛)

The heat equation is linear, and so by the method of superposition, the most general solution, where

all positible terms as a linaer combination is:

𝑢(𝑥, 𝑡) = ∑ 𝐴𝑛 sin (𝑛𝜋

𝐿𝑥) 𝑒

−𝑛2𝜋2

𝐿2 𝜅𝑡

∞

𝑛=1

The Initial conditions: 𝑢(𝑥, 0) = 𝑓(𝑥)

⟹ ∑ 𝐴𝑛 sin (𝑛𝜋

𝐿𝑥)

∞

𝑛=1

= 𝑓(𝑥)