Video # |
Video Tutorial Title |
Remarks |

1 | Classification of Differential Equations | Discussing all the terms by which we classify differential equations |

2 | What are Power Series Used For? | I try to motivate the practical use of power series in physics |

3 | Where Power Series Start | Do they start at n=0, n=1, n=3 etc? |

4 | Shifting Power Series Indices | A very important skill; shifting from n=1 to n=0 (say) |

5 | Differentiating Power Series | As simple as it sounds but important when we discuss the method of Frobenius |

6 | Multiplying Power Series by Functions | Discussing the impact on where the power series start |

7 | The Characteristic Equation 1/2 | Deriving the Characteristic Equation for solving equations with constant coefficients |

8 | Introduction to Power Series Solutions | How differential equations are solved using the method of power series solutions |

9 | General and Particular Solutions | Not yet recorded |

10 | The General Solution | To 2nd order D.Es., with constant coefficients using power series |

11 | Power Series Example 1 | y”- y’ = 0 |

12 | Power Series Example 2 | y” + 4y = 0 |

13 | Power Series Example 3 | y” — 9y = 0 |

14 a | Proof of the Wronskian | Find out if your solutions are linearly dependent or not |

14 b | Introduction to the Method of Frobenius | Solving equations which are not analytic using power series |

15 a | Example 1 Part 1 | x^{2} + y” + (x — x^{2})y’ — y =0 |

15 b | Example 2 Part 2 | x^{2} + y” + (x — x^{2})y’ — y =0 |

16 a | Example 1 Part 1 | xy” + 2y’ + xy = 0 |

16 b | Example 1 Part 2 | xy” + 2y’ + xy = 0 |

17 | Example 3 | (x^{2} + 1)y” + xy’ — y = 0 |

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20 | Introduction to Laplace’s Equation | V” = 0 |

21 | Earnshaw’s Theorem | No local maxima or minima are permitted |

22 | First Uniqueness Theorem | Why the function’s value at the boundary is important |

23 | Second Uniqueness Theorem | |

24 | Introduction to Separation of Variables | The fundamentals of using this very powerful technique to solve differential equations |

25 | Characteristic Equation 2/2 | Derivation of the Characteristic Equation |

26 | Separation of Variables Example 1 | V” = 0 to include the use of Fourier Series |

27 | Separation of Variables Example 2 | V” = 0 (new boundary conditions) to include the use of Fourier Series |

28 | Separation of Variables Example 3 | V” = 0 (new boundary conditions) to include the use of Fourier Series |

29 | Properties of Separable Solutions | Why stationary state solutions are the fundamentals of Quantum Mechanics |

30 | Spherical Polar Co-ordinates | Rectangular to spherical polar co-ordinates is required for the wavefunctions of Hydrogen |

31 | Laplace Equation in Spherical Co-ords | Converting the D.E., to spherical polar coordinates |

32 | The Radial Equation | The first of the three ordinary equations resulting from the method of separation of variables |

33 | Laplacian in Spherical Coordinates | A very tedious derivation of a very important operator |

34 | The Azimuthal Equation | The second of the three ordinary equations resulting from the method of separation of variables |

35 | The Polar Angle Equation | The third of the three ordinary equations resulting from the method of separation of variables |

36 | Schrodinger Equation in Spherical Co-ords | Simplifying the derivation of the Hydrogen atom wavefunctions |

37 | The Radial Equation for Hydrogen | |

38 | Wavefunctions of Hydrogen | A derivation of the wavefunctions of the Hydrogen atom |

39 | Quantum Numbers | Where they come from and why they are useful |

40 | Electron Orbitals | The Periodic Table |

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