Math 2.0

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Ordinary differential equations

108 ideas in teaching order. Each idea comes after the ideas that it builds on.

Start with Differential equation →

  1. Differential equation concept
  2. Ordinary differential equation definition
  3. Order of a differential equation definition
  4. Solution of an ordinary differential equation definition
  5. Explicit and implicit differential equations concept
  6. General solution and particular solution concept
  7. Initial value problem definition
  8. Slope field definition
  9. Reduction of a higher-order equation to a first-order system statement
  10. Separable equation definition
  11. Exponential growth and decay example
  12. Newton's law of cooling example
  13. Logistic equation example
  14. Linear first-order equation definition
  15. Integrating factor definition
  16. Solution formula for linear first-order equations statement
  17. Homogeneous first-order equation definition
  18. Exact equation definition
  19. Test for exactness statement
  20. Integrating factor for a non-exact equation concept
  21. Bernoulli equation definition
  22. Riccati equation definition
  23. Clairaut's equation example
  24. Singular solution definition
  25. Linear differential equation of order n definition
  26. Homogeneous linear equation definition
  27. Superposition principle statement
  28. Solution space of a homogeneous linear equation has dimension n statement
  29. Wronskian definition
  30. Wronskian test for linear independence of solutions statement
  31. Abel's identity statement
  32. Fundamental system of solutions definition
  33. General solution of an inhomogeneous linear equation statement
  34. Linear equation with constant coefficients definition
  35. Characteristic equation definition
  36. Solutions for distinct real roots statement
  37. Solutions for repeated roots statement
  38. Solutions for complex conjugate roots statement
  39. Cauchy–Euler equation definition
  40. Reduction of order concept
  41. Method of undetermined coefficients concept
  42. Variation of parameters concept
  43. Harmonic oscillator example
  44. Damped harmonic oscillator example
  45. Underdamped concept
  46. Forced oscillation and resonance example
  47. Power series solution concept
  48. System of linear differential equations definition
  49. Homogeneous linear system with constant coefficients definition
  50. Eigenvalue method for linear systems concept
  51. Matrix exponential as the solution of a linear system statement
  52. Fundamental matrix definition
  53. Liouville's formula statement
  54. Variation of constants formula for linear systems statement
  55. Lipschitz condition definition
  56. Picard–Lindelöf hypotheses definition
  57. C^1 vector fields satisfy the Picard–Lindelöf hypotheses statement
  58. Picard operator definition
  59. Integral equation form of an initial value problem statement
  60. Picard iteration concept
  61. An iterate of the Picard operator is a contraction statement
  62. Picard–Lindelöf theorem statement
  63. Picard–Lindelöf theorem in integral form statement
  64. Uniqueness of solutions for Lipschitz vector fields statement
  65. Non-uniqueness for a non-Lipschitz equation counterexample
  66. Peano existence theorem statement
  67. Grönwall's inequality statement
  68. Grönwall bound definition
  69. Discrete Grönwall inequality statement
  70. Continuous dependence on initial data statement
  71. Estimate for approximate solutions statement
  72. Maximal solution definition
  73. Blow-up in finite time example
  74. Global existence for globally Lipschitz vector fields statement
  75. Autonomous system definition
  76. Time translation of solutions of an autonomous system statement
  77. Flow of a vector field definition
  78. Trajectory definition
  79. Equilibrium point definition
  80. Phase line concept
  81. Phase plane concept
  82. Phase portrait concept
  83. Nullcline definition
  84. First integral definition
  85. Hamiltonian system definition
  86. Periodic orbit definition
  87. Limit cycle definition
  88. Poincaré–Bendixson theorem statement
  89. Lyapunov stability definition
  90. Asymptotic stability definition
  91. Stability of linear systems via eigenvalues statement
  92. Classification of equilibria of planar linear systems concept
  93. Linearisation at an equilibrium concept
  94. Hyperbolic equilibrium point definition
  95. Hartman–Grobman theorem statement
  96. Lyapunov function definition
  97. Lyapunov's stability theorem statement
  98. LaSalle's invariance principle statement
  99. Lotka–Volterra equations example
  100. Pendulum equation example
  101. Laplace transform definition
  102. Laplace transform of a derivative statement
  103. Inverse Laplace transform definition
  104. Heaviside step function object
  105. Dirac delta function object
  106. Convolution theorem for the Laplace transform statement
  107. Solving initial value problems with the Laplace transform concept
  108. Euler method concept