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 ------------------------------------------------------------ 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) ------------------------------------------------------------ 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 ------------------------------------------------------------ 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 ------------------------------------------------------------ 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) ------------------------------------------------------------ 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] ------------------------------------------------------------ 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 ------------------------------------------------------------ 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 ------------------------------------------------------------ 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 ------------------------------------------------------------ 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) ------------------------------------------------------------ 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 ------------------------------------------------------------ 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 (**)