dt amount entering amount leaving This results in the DEs describing the

Dt amount entering amount leaving this results in the

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dt = amount entering - amount leaving This results in the DEs describing the amounts dA 1 dt = f 1 q 1 + f 3 c 2 - f 4 c 1 dA 2 dt = f 2 q 2 + f 5 c 1 - f 3 c 2 These are transformed into concentration equations by dividing by V 1 and V 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (6/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example Basic Mixing Problem 4 Concentration Equations dc 1 dt = f 1 q 1 + f 3 c 2 V 1 - f 4 V 1 c 1 dc 2 dt = f 2 q 2 + f 5 c 1 V 2 - f 3 V 2 c 2 This can be written as a system of 1 st order linear DEs ˙ c 1 ˙ c 2 = - f 4 V 1 f 3 V 1 f 5 V 2 - f 3 V 2 c 1 c 2 + f 1 q 1 V 1 f 2 q 2 V 2 with c 1 (0) = c 10 and c 2 (0) = c 20 , which in shorthand is ˙ c = Ac + Q Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (7/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example Basic Mixing Problem 5 Equilibrium: We find the equilibrium by solving Ac e = - Q or - f 4 V 1 f 3 V 1 f 5 V 2 - f 3 V 2 c 1 e c 2 e = - f 1 q 1 V 1 - f 2 q 2 V 2 This has the general solution c 1 e c 2 e = f 1 q 1 + f 2 q 2 f 4 - f 5 f 1 f 5 q 1 + f 2 f 4 q 2 f 3 ( f 4 - f 5 ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (8/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example Basic Mixing Problem 5 Eigenvalues: We find the eigenvalues by solving det | A - λ I | = 0 or det - f 4 V 1 - λ f 3 V 1 f 5 V 2 - f 3 V 2 - λ = 0 This has the characteristic equation λ 2 + f 4 V 1 + f 3 V 2 λ + f 3 ( f 4 - f 5 ) V 1 V 2 = 0 Since det | A | > 0, discriminant D > 0, and tr ( A ) < 0, the Stability Diagram from before shows this system has a Stable node or sink , as we would expect Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (9/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts Mixing Problem Example Pharmokinetic Problem LSD Example Mixing Problem Example 1 Mixing Problem Example Assume the following parameters: V 1 = 100 l, V 2 = 60 l, q 1 = 7 g/l, q 2 = 12 g/l, f 1 = 0 . 2 l/min, f 2 = 0 . 15 l/min, f 3 = 0 . 25 l/min, f 4 = 0 . 45 l/min, f 5 = 0 . 1 l/min, f 6 = 0 . 35 l/min This can be written as a system of 1 st order linear DEs ˙ c 1 ˙ c 2 = - 0 . 0045 0 . 0025 0 . 00167 - 0 . 004167 c 1 c 2 + 0 . 014 0 . 03 with c 1 (0) = 2 g/l and c 2 (0) = 1 g/l Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (10/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Basic Mixing Problem - Water and Inert Salts
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