# models - M224-228 Modeling David Gurarie d All DE models...

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M224-228: Modeling David Gurarie January 19, 2001 All DE models have form: d dt [ state ]= dynamic law [ state; t;::: ] . Solutions of such systems are certain functions of time t , that also depend on many other pa- rameters (initial state, coe¢cients of the equation, etc.). We list a few examples and show typical solution curves as functions of t . 1P o p u l a t i o n g r o w t h 1. Linear: y 0 = ky ; y ( t ) - population, k -growthrate , [ k 1 sec 0 2 4 6 8 10 0.5 1 1.5 2 2.5 3 t 2. Logistic: y 0 = k (1 ¡ y=N ) y ; N - carrying capacity 5 10 15 20 25 30 t 0.5 1 1.5 2 2.5 3 3.5 4 Pop 1

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2 Chemical reaction Reaction: A + B k ! AB , x;y -concentrations (per unit volume [ x ]= gr cm 3 )o f A , B obey a coupled di¤erential system of equations: ½ x 0 = ¡ kxy y 0 = ¡ kxy ; k - reaction rate, [ k cm 3 sec . Conserve di¤erence of two concentrations: d dt ( x ¡ y )=0 , hence reduce to a single DE: y 0 = ¡ ky ( y + C ) , C = x ¡ y -constant 3 Mechanics 3.1 Oscillator (mass-spring) Newton’s law: m Ä x = ¡ kx + f -2-nd order DE in x ( t ) -position; m -mass, k -spring constant, f -external force. Newton’s law could be written as di¤erential
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models - M224-228 Modeling David Gurarie d All DE models...

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