MTH401 Handouts Set 2 by VUBuddy

Table of Contents

Introduction ................................................................................................................. 1 Fundamentals .............................................................................................................. 3

2. 1 Elements of the Theory ........................................................................................ 3
3. 2 Specific Examples of ODE’s ............................................................................... 3
4. 3 The order of an equation ...................................................................................... 4
5. 4 Ordinary Differential Equation ............................................................................ 4
6. 5 Partial Differential Equation ................................................................................ 4
7. 6 Results from ODE data ........................................................................................ 4
8. 7 BVP Examples ..................................................................................................... 5
9. 8 Properties of ODE’s ............................................................................................. 5
10. 9 Superposition ........................................................................................................ 5
11. 10 Explicit Solution ............................................................................................... 5
12. 11 Implicit Solution ............................................................................................... 5 Separable Equations .................................................................................................... 6
13. 1 Solution steps of Separable Equations ................................................................. 6
14. 2 Exercise .............................................................................................................. 13 Homogeneous Differential Equations ....................................................................... 15
15. 1
16. 2 Equations reducible to homogenous form ......................................................... 17
17. 2.1 Case 1 ......................................................................................................... 18
18. 2.2 Case 2 ......................................................................................................... 18
19. 3 Exercise .............................................................................................................. 20 Exact Differential Equations .................................................................................... 22
20. 1
21. 2 Exercise .............................................................................................................. 26 Integrating Factor Technique .................................................................................... 28
22. 1 Case 1 ................................................................................................................ 28
23. 2 Case 2 ................................................................................................................ 29
24. 3 Case 3 ................................................................................................................ 29
25. 4 Case 4 ................................................................................................................. 29
26. 5 Exercise .............................................................................................................. 36 First Order Linear Equations .................................................................................... 37
27. 1
28. 2 Exercise .............................................................................................................. 41 Bernoulli Equations .................................................................................................. 43
29. 1
30. 2 Exercise .............................................................................................................. 47
31. 3 Substitutions ....................................................................................................... 48
32. 4 Exercise .............................................................................................................. 52 Solved Problems ....................................................................................................... 53 Applications of First Order Differential Equations ............................................... 69
33. 1 Orthogonal Trajectories ................................................................................. 69
34. 2 Orthogonal curves .......................................................................................... 71
35. 3 Orthogonal Trajectories (OT) ........................................................................ 71
36. 3.1 Method of finding Orthogonal Trajectory .............................................. 72
37. 4 Population Dynamics ..................................................................................... 77 Radioactive Decay ................................................................................................. 80
38. 1 Newton's Law of Cooling ............................................................................... 82
39. 2 Carbon Dating ................................................................................................ 84 Applications of Non-linear Equations ................................................................... 86
40. 1 Logistic equation ............................................................................................ 86
41. 1.1 Solution of the Logistic equation ................................................................ 86
42. 1.2 Special Cases of Logistic Equation............................................................. 87
43. 1.3 A Modification of LE.................................................................................. 88
44. 2 Chemical reactions ......................................................................................... 89
45. 3 Miscellaneous Applications ............................................................................ 92 13 Higher Order Linear Differential Equations ............................................................. 94
46. 1
47. 2 Initial -Value Problem .................................................................................... 94
48. 2.1 Solution of IVP ........................................................................................... 95
49. 3 Theorem ( Existence and Uniqueness of Solutions) ....................................... 95
50. 4 Boundary-value problem (BVP) ..................................................................... 97
51. 4.1 Solution of BVP .......................................................................................... 97
52. 4.2 Possible Boundary Conditions .................................................................... 98
53. 5 Linear Dependence ....................................................................................... 100
54. 6 Linear Independence ..................................................................................... 100
55. 6.1 Case of two functions................................................................................ 100
56. 7 Wronskian ..................................................................................................... 102
57. 8 Theorem (Criterion for Linearly Independent Functions) ............................ 102
58. 9 Exercise ........................................................................................................ 104 14 Solutions of Higher Order Linear Equations .......................................................... 106
59. 1
60. 2 Superposition Principle................................................................................. 106
61. 3 Linear Independence of Solutions ................................................................ 109
62. 4 Fundamental Set of Solutions ....................................................................... 109
63. 4.1 Existence of a Fundamental Set ................................................................ 110
64. 5 General Solution-Homogeneous Equations.................................................. 110
65. 6 Non-Homogeneous Equations ...................................................................... 112
66. 7 Complementary Function ............................................................................. 112
67. 8 General Solution of Non-Homogeneous Equations ..................................... 113
68. 9 Superposition Principle for Non-homogeneous Equations .......................... 114
69. 10 Exercise ........................................................................................................ 115 15 Construction of a Second Solution ......................................................................... 117
70. 1 General Case ................................................................................................. 117
71. 2 Order Reduction ........................................................................................... 120
72. 3 Exercise ........................................................................................................ 123 16 Homogeneous Linear Equations with Constant Coefficients ................................. 124
73. 1
74. 1.1 Case 1 (Distinct Real Roots) ..................................................................... 124
75. 1.2 Case 2 (Repeated Roots) ........................................................................... 125
76. 1.3 Case 3 (Complex Roots) ........................................................................... 125
77. 2 Higher Order Equations ................................................................................ 127
78. 2.1 Case 1 (Real distinct roots) ....................................................................... 127