The references refer to the books in the list of text books linked from the sidebar.
Laplace Transforms
Lecture 1: Introduction. Laplace transform: definition. The Laplace transform of constants and exponentials. Linearity and the example of hyperbolic and trignometric functions. (K 251-253).
Lecture 2: Laplace transform of powers of t. The first shift theorem. (K 253). The Heaviside function and its Laplace transform (K 265-266). The Laplace transform of the differential of a function (K 258).
Lecture 3: A hard example of using the shift theorem (note 2). The Laplace transform of the f'' (K 259). Examples of using the Laplace transform to solve differential equations. An example with partial fractions. (K 285).
Lecture 4: Lots of examples of solving differential equations using Laplace transforms. The partial fraction expansion when there is a repeated root. (K 285 again).
Lecture 5: Another example with a repeated root. An example with complex roots. (see note 4)
Lecture 6: Another example with complex roots. General discussion of differential equations and some terminology used. (K 260).
Lecture 7: The second shift theorem (K 267), examples of using the 2nd shift theorem. Examples of solving differential equations where the second shift theorem is needed, (note 5).
Lecture 8: The Dirac delta functions (K 270), example with the Dirac delta function.
Lecture 9: The convolution theorem (K 279-280), some convolutions (see note 6 for an example). Begining of periodic functions.
Lecture 10: Revision of convolutions (note 7). Laplace transforms of periodic functions (K290, note 8 and note 9).
Z-Transforms
Lecture 11: Review of sequences and series. The Z-tranforms (J
219). The Z-transform of a geometric sequence, differenciating this
formula (J 212). (see also note 12).
Lecture 12: The Z-tranforms and the Z-transform of a geometric
sequence again. The Z-transform of the unit pulse (1,0,0,...). Also,
properties of the Z-transform. The shift theorems, delaying (J 225)
and advancing (J 227). (see also note 13).
Lecture 13: cancelled.
Lecture 14: More on the advancing theorem. Inverting
Z-transforms (J 230-232). Using the Z-transform to solve difference
equations (J 239).
Lecture 15: Using the Z-tranform to solve difference equations, some examples, example with repeated root, example with unit pulse.
Lecture 16: Using the Z-tranform to solve difference equations, example with unit pulse, example with less convenient initial conditions.
Systems of linear differential equations
Lecture 17: Begining systems of differential equations. (K chapter 3).
Lecture 18: Christmas Quiz, with spot prizes (note 17).
Lecture 19: The mixing problem (K 152-154). A summary of how to solve a linear system.
Lecture 20: Linear homogenous first order equations, definitions and discussion (K 160), some examples. Begining of phase portraits.(K 162-167, note 20).
Lecture 21: Solving systems of equations and drawing the phase portrait, a saddle point example.
Lecture 22: Solving systems of equations and drawing the phase portrait, an improper node example. (note 19)
Lecture 23: More nodes, start of circles.
Lecture 24: Circle nodes (K 162-167, note 20).
Lecture 25: Spirals. Only one eigenvector (K 167-169).
Lecture 26: More on only one eigenvector, begining inhomogeneous
system: revising a single inhomogenous equation (note 21).
Lecture 27: Inhomogeneous equations again, a system of
inhomogeneous equations (K 187, note 14).
Lecture 28: More inhomogenous equation examples. Converting a
second order equation to first order (K 156-157, note 22). Start of
linearization (K176-177).
Linearization
Lecture 29: Linearization, the pendulum. Start another example.
Lecture 30: Finish previous example (schol. 2001). Wind resistence, spirals.
Series solution
Lecture 32: A bit more on linearization with spirals. Series solutions, general explaination and simple example (K 194-205).
Lecture 33: Series solutions, more examples.
Lecture 34: Series solutions, more examples, method of Frobenius.
Lecture 35: Series solutions, method of Frobenius (K 211-212, note 26 ). Bessel's eqn (schol. 2002).
Lecture 36: The Legendre equation, Legendre polynomials (K 205).
Vector calculus
Lecture 37: Revision of vectors, dot products, cross products (K 400-422), start of curves in space.
Lecture 38:: Length of a curve (K 432), introduction to scalar and vector fields, the gradient.
Lecture 39: Gradient of a scalar field, the directional derivative (K 446-450), normal to a surface.
Lecture 40: More on the normal, start of div. (K 453-456).
Lecture 41: More on div, continuity equation (K 454-455). Curl. (K 457-459)
Lecture 42: Good friday, no lecture.
Lecture 43: More on curl. Start of the Gauss theorem (K 505-509).
Lecture 44: More on Gauss theorem, mention of the heat equation.
The Heat Equation
Lecture 45: The Heat equation, solving it (K 600-602).
Lecture 46: Solving the heat equation, continued.
Lecture 47: Different boundary conditions. (K 604, note 27)
Lecture 48: Revision of heat equation. Concluding remarks.