Practicum 1 - February 17

Introduction: ODE -- definition, concept of solution

ODE type: with separated variables
demo:       x'=3t², x'=5x , x' = -t/x
class:      x'=x²+1, x'=e^x(t+1) , x'=(x^2-x)/t

Theorem P.1 - sketch of the proof

demo: 	x'=2√|x|.exp(-t)	(only for x>0)

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Practicum 2&3 - February 24

finishing x'=2√|x|.exp(-t) (for x=0 and x<0)

ODE type: first order linear

class:  x'-x=te^t
        x'-x/t = t^2e^t
        x-x/t^2 = 1/t^3
        x'+2x=cos t

demo:   tx'-3x=t³ + initial condition: x(1)=-1
		+ remark on maximal solutions

ODE type: homogeneous

demo:	x'=(x-t)/(x+t) ... (can't be solved)

        1.  x'=exp(x/t)+x/t
        5.  t²x' = x² + 2tx
        6.  2txx' + t²-x² = 0


ODE type: Bernoulli

demo:	ex.6) above via z=x²

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Practicum 4 - March 3 

topic: EQA (elementary qualitative analysis)

demo:	x'=x²+t²-1

class: 
        1. x'=t²(x+1)
        2. x'=(x-1)/(t-1)
        3. x'=t(x+1)
        4. x'=x/t+t²
        5. x'=2tx-2

theory:		Peano & Picard, 
			symmetries: proof for f(-t,-x)=f(t,x)

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Practicum 5 - March 10 

topic: EQA 2

demo:   x'=3y² , y'=-2x     

class:  0. x'=y , y'=-x
        1. x'=x(1-x)-xy , y'=-2y+xy
        2. x'=x(1-x/2-y) , y'=y(2-2x-y)
        3. x'=x(2-2x-y) , y'=y(1-x/2-y)

Note:	1--3 in 1st quadrant only (x,y>0)

on board: 	no. 0 (prime integral preview)
			no. 1 (demo by student)

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Practicum 6 - March 17

Review of theory:
	- general system of ODEs, Peano & Picard (Thm P.4 & P.5)
	- prime integral: definition, characterization (Thm P.7)

demo:	x'=3y² , y'=-2x 	(prime integral: y³+x²=c)

class: find prime integral(s) for the systems:

		1. x'=x, y'=-y
		2. x'=xy, y'=xy
		3. x'=y, y'=x-x² (or x"+x²-x=0)
		4. x'=xy, y'=xz, z'=yz

theory:	stability, asymptotic stability, instability
		Theorem P.5 [Linearized (in)stability theorem]

application: example above ... conclusion?

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Practicum 7 - March 24

Recall (informally) concept(s) of stability
Examples (by prime integrals): 
		x'=x, y'=-y	(stable, not asymptotically)
		y³+x²=c		(unstable)

Linearization: general motivation
Theorem of linearized (instability)

class:	1. x'=x(2-x-y), y'=y(x-1)	
		2. x'=x(1-x-y/(x+1/4)), y'=y(1-4y/3x)	{H-T}

Ad 1)	E₁=(1,1)	... stable spiral
		E₂=(2,0)	unstable direction v=(1,-3/2) 

Theorem P.6 [Stable/unstable direction]
		... see example E₂=(2,0) above ...

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Practicum 8 - March 31

Recall  (1) X'=F(X) and its linearization
		(2) U'=AU

Theorems P.5,P.6 and Hartman-Grobman
		... motivation to study (2)

Some (informal) remarks on incomplete (yet) theory to (2):
	∙	matrix exponential
	∙	solution(s) in the form exp(λt)v
	∙	reformulate as one equation of order n
		(example: A=[0 1;4 0], x=exp(±2t)

Class - for team 1 - 5, let A=
	1)	diag(1,2)
	2)	diag(-2,1)
	3)	[-2,1;0,-2]
	4)	[-1,-2;2,-1]
	5)	[0,1;0,0]
	
(On board, solutions to 1,2,4 were presented.)
Preliminary info on the "final project".

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Practicum 9 - April 7

(Problems 3) and 5) from the last class.)

Theorem P.8 -- on solutions to (L-1) and (L-2)
matrix exponential: some properties
Example: exp(tA₃) = diagonal + nilpotent
linear n-th order with const. coeff.: Ansatz -> characteristic poly

Class - matrix exponential:
	1)	[ a 0 ; b 0 ]
	2)  [ 0 1 10 ; 0 0 -1 ; 0 0 0]
	3)  [ α -β ; β α ]
		characteristic poly + general solution:
	4)  x"" + 16x = 0
	5)  x"" = 0
	6)  x"+3x'-40x=0
	7)  x'''-3x"+3x'-x=0

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Practicum 10 - April 14

topic: stability part 2 - Lyapunov functions

motivation: HW3.3

Definition: (strict) Lyapunov function
Theorem P.9 [Lyapunov theorem]

Example:	pendulum (with friction)

Class - L.f. & stability:

	0)	verify that \dot{V}_F \le 0 for the pendulum above
	1)	x'=-2y-x³ , y'=3x-y³
	2)	x'=-x/2-y² , y'= xy - 7x²y
	3)	x'=-2y³ , y'=x
	4)	x'=-y+2x³ , y'=2x+y³

Hint: V=ax²+by² with a,b>0; more generally ax^{2n}+by^{2m}, m,n natural numbers


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Practicum 11 - April 28

topic: numerical methods

1) Euler's method + error estimate
- class problems: solve (exactly) / code Euler 
2) method of curvature / osculating circle

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Practicum 13 - May 12 

Final project - presentations part I

HW 1.1, HW 1.3, HW 2.3, matrix (symbolic) calculus

Remarks on number classes (rational, algebraic, constructible, computable)
Impossibility of (some) Euclidean construction(s).

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Practicum 14 - May 19 

HW 1.2, HW 2.3, orbital derivative (symbolic)