Recordings of practice classes for DEN(E) (2021)

These are recordings of practice classes taught online during the covid locdown re-edited to better fit the way we teach this course now. It is not a perfect fit, but the substance is there.

Week 01: Solving ODEs of order 1 using separation
0:00:00 first problem (with logarithmic trick) - separation
0:15:14 first problem - initial conditions
0:22:43 first problem - asymptotic growth
0:26:52 second problem (with square root) - separation and initial conditions
0:47:42 third problem with interesting features
(1:09)
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Week 02: Variation of parameter, analyzing behaviour of solutions
0:00:00 are the two problems from previous class linear?
0:03:23 problem on variation
0:20:02 first problem for analysis (slope field for nonseparable, stationary solutions)
0:43:27 second problem (slope field, stationary solutions, stability)
1:03:06 third problem (slope field for separable, stationary solutions)
(1:19)
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Week 03: Numerical mathematics - error, derivation, integration
0:00:00 approximating a function, the O(h^p) notation
0:28:31 propagation of error in cosine (theoretically and practically)
0:41:15 approximating derivative by differences (by hand, numerical experiments)
0:51:30 approximating integral (by hand for exam, numerical experiments)
1:08:13 working with error estimate
(1:34)
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Week 04: Solving ODEs of order 1 numerically (Euler)
0:00:00 theoretical question on the Euler method
0:03:56 calculating (by hand) question on the Euler method
0:12:25 more advanced methods, playing with Maple
0:23:19 working with the error estimate
0:34:25 playing with estimate of method order
0:40:28 another problem on the Euler method
0:47:17 problem on predicting error
(0:52)
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Week 05: Solving linear differential homogeneous equations
0:00:00 four small problems that cover perhaps all possible cases
0:26:52 an example with initial conditions
0:34:41 some non-routine questions
0:53:18 an example with a parameter
1:16:16 review of linear equations, return to variation
(1:26)
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Week 06: Solving linear differential (nonhomogeneous) equations
0:00:00 introductory example
0:20:06 practicing the guessing stage using a chart
0:30:01 an exam-level example
0:47:12 example: equation of order 3
0:55:16 practice miniproblems with behaviour at infinity
1:06:57 bonus problem
(1:21)
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Week 07: Finding roots/solving equations numerically (classical methods)
0:00:00 the main example
0:31:13 second example (practice problem)
0:44:40 third example, stopping conditions, tests in Maple
1:12:50 bonus: example in Maple, general strategy
(1:20)
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Week 08: Finding roots/solving equations numerically (fixed point)
0:00:00 the main example on calculations by hand
       (14:13 teoretical sidestep: the fixed point theorem)
0:24:25 relaxation for the first example
0:37:12 second example
0:50:43 bonus example, playing with Maple
(1:35)
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Week 09: Solving homogeneous systems of linear differential equations
0:00:00 a typical example for exam
       (17:33 bonus on stationary solutions and stability)
0:30:49 an example with complex eigenvalues
0:44:17 practice problem
0:51:02 transforming a higher-order equation into a system
0:55:15 a problem to solve
(1:01)
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Week 10: Solving nonhomogeneous systems of linear differential equations
0:00:00 an example, solved by guessing
0:20:22 the same example solved by variation
(0:38)
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Week 11: Solving systems of linear equations by elimination
0:00:00 introductory example of exam type
0:32:13 practice problem
0:39:12 the same system, but with a new right-hand side
0:46:26 inspiration for LU-decomposition and how it works
1:00:48 recalling spectral radius and matrix norms
1:01:11 playing with Maple
1:21:25 bonus: another practice problem with an interesting alternative
1:25:29 that alternative: fraction-free elimination
1:29:56 practice problem on fraction-free elimination
1:35:06 question about applications with huge matrices
(1:40)
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Week 12: Solving systems of linear equations by iteration
0:00:00 introduxctory example of exam type
0:19:40 second example
0:29:59 third example (actually the first one rearranged)
0:35:30 relaxation
0:38:33 a couple of sad news about matrix iteration
0:42:44 playing with Maple
1:03:50 practice problem
(1:17)
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Week 13: Numerical addenda
0:00:00 finding eigenvalues and eigenvectors numerically
0:16:20 solving systems of ODEs numerically: the Euler method
(0:34)
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Week 13x: Special: Practice final
0:00:00 Sample final test
(0:38)