Practicum 1&2 - February 19

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'=x²+1 -> interval discussion

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ODE type: 1st order linear
Theorem P.2 + sketch of the proof
demo:   x'+(cos t)x = (1+t)e^(-sin t)

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

ODE type: Bernoulli - solve by substitution z=x^(1-α)
demo:	 x' -4x/t = t√x
		discussion of the intervals!! (x>0 ⇒ z>0)


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Practicum 3 - March 4

demo:	x'=2√|x|.exp(-t)
		x'-tx=-x³exp(-t²)

class:	(and demo no. 3)
        1.  x'=exp(x/t)+x/t
        2.  x'-2tx = 2t³x²
        3.  x' = (x-t)/(x+t)
        4.  x' - 9t²x = (t⁵+t²)\root{3}\of{x²}
        5.  t²x' = x² + 2tx
        6.  2txx' + t²-x² = 0


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Practicum 4 - March 11

EQA = elementary qualitative analysis

demo:	x'=e^x(t+1), x'=2√|x|e^{-t}

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

Theorem P.3 & P.4 [Peano & Picard]
Remark on solution symmetries 

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Practicum 5 - March 18

EQA = elementary qualitative analysis [2]


demo:	x'=2tx-2; (+ symmetry w.r. to origin)

demo:	x"+x=0 and x'=x²-2x+y , y'=y(1-x)

class:
        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)

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Practicum 6 - March 25

demo:	x'=3y² , y'=-2x 	(1st integral: y³+x²=c)

class: find 1st integral(s) for the systems:

		1. x'=y, y'=-x
		2. x'=y, y'=x-x²
		3. x'=xy, y'=xz, z'=yz

theory:	stability, asymptotic stability, instability
		linearization principle
		solution of u'=Au via eigenvalues/eigenvectors
		Theorem P.5 [Linearized (in)stability theorem]

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Practicum 7 - April 8 

theory: matrix of linearization, Theorem P.5 and P.5
			[linearized (in)stability, (un)stable direction]

demo:	HW6 x'=x(2-x-y), y'=x(1-y)

theory:	first integral - definition, Theorem P.7
				[characterization of 1st integral]

demo:	x'=-y,	y'=x
	x'=x²y,	y'=xy²

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Practicum 8 - April 15 

demo:	x''+sin(x)=0

class:	x'=-2x, y'=-y
	x'=2x, y'=-y
	x'=-2x, y'=-2y
	x'=-2y, y'=2x

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Practicum 9 - April 22 

theory:		linear homogeneous ODEs with constant coeffs.
			(α) system of 1st order equations
			(β) one equation of higher order

class:		
1.	x""-3x"+2x=0
2.	x'''+3x''+3x'+x=0
3.	x""+18x"+81x=0
4.	x'=10x-6y, y'=18x-11y
5.	x'=y, y'=z, z'=y

simpler problems: x"+2x'-3x=0, x"+x=0, x"+2x'+x=0

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Practicum 10 - April 29 

theory:		characteristic polynomial -> fundamental system
			note on initial conditions
demo:		x"+x=0, x"+2x'+x=0

theory:		hyperbolic points / matrices
			Hartman-Grobman theorem

			classification of 2x2 hyperbolic matrices:
			1) node 2) saddle point 3) comples roots
			... discussion of cases 1) and 2)

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Practicum 11 - May 6 

project presentations: HW1, HW2, HW4.1, HW5, HW7.1

demo:	Euler's (1st order) method
		curvature method (2nd order)

class:	matrix exponentials for A=
		(1) [0 1; 0 0]
		(2) diag(-1 2)
		(3) [1 -2 ;2 1]
		(4) [a -b ; +b a]
		
theory:	notions of stability,
		(counter)example to Theorem P.5:
		x'=-2y³, y'=x

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Practicum 12 - May 20 


Practicum 11 - May 6 

Lyapunov (strict) function

examples:	pendulum, Prac 11/(1)

class:	x'=-y-x³,	y'=x-y³
		x'=-x-y²,	y'=xy-x²y
		y'=2y+x³,	y'=-x+y³	(**)