Lecture40 - System of ODEs ❚ A system of simultaneous...

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Unformatted text preview: System of ODEs ❚ A system of simultaneous ODEs ❚ n equations with n initial conditions = = = ) , , , , ( ) , , , , ( ) , , , , ( n 2 1 n n n 2 1 2 2 n 2 1 1 1 y y y t f dt dy y y y t f dt dy y y y t f dt dy System of ODEs ❚ Bungee Jumper’s velocity and position ❚ Two simultaneous ODEs v x v m c g dt dv v dt dx 2 d = = - = = ) ( ) ( = = t m gc c m t x t m gc c gm t v d d d d cosh ln ) ( tanh ) ( Solution: Second-Order ODE ❚ Convert to two first-order ODEs = = = 1 2 2 ) (t dt dy t y dt dy y t g dt y d α α , ) ( ) , , ( = = = = = = ⇒ = = 1 2 1 2 1 2 2 2 2 1 2 1 t y t y s C I y y t g dt y d dt dy y dt dy dt dy dt dy y y y let α α ) ( ) ( . . ) , , ( System of Two first-order ODEs Euler’s Method ❚ Any method considered earlier can be used ❚ Euler’s method for two ODE-IVPs ❚ Basic Euler method ❚ Two ODE-IVPs y y t hf y y y y t hf y y i 2 i 1 i 2 i 2 1 i 2 i 2 i 1 i 1 i 1 1 i 1 + = + = + + ) , , ( ) , , ( , , , , , , , , ) , ( i i i 1 i y t hf y y + = + y 1 y 2 y 3 1 2 t Hand Calculations: Euler’s Method ❚ Solve the following ODE from t = 0 to t = 1 with h = 0.5 ❚ Euler method [ ] [ ] [ ] 1, 1 1, 1 1, 2, 1, 1, 2, 1 2, 2 1, 2, 2, 1, 2, 1 1 1 2 2 1 2 ( , , ) (0.5 ) ( , , ) (4 0.1 0.3 ) y (0.5) y (0) 0.5y (0) 4 ( 0.5)(4) (0.5) 3 y (0.5) y (0) 4 0.1y (0) 0.3 (0) 6 4 0.1(4) 0. i i i i i i i i i i i i i i i y y f t y y h y y h y y f t y y h y y y h h y h + + = + =- = + = +-- = +- = +- = = +-- = +-- [ ] [ ] [ ] [ ] [ ] 1 1 1 2 2 1 2 3(6) (0.5) 6.9 y (1.0) y (0.5) 0.5y (0.5) 3 ( 0.5)(3) (0.5) 2.25 y (1.0) y (0.5) 4 0.1y (0.5) 0.3y (0.5) 6.9 4 0.1(3) 0.3(6.9) (0.5) 7.715 h h = = +- = +- = = +-- = +-- = = = -- =- = 6 y 4 y y 3 y 1 4 dt dy y 5 dt dy 2 1 2 1 2 1 1 ) ( ) ( . . . Euler’s Method for a System of ODEs y is a column vector with n variables function f = example5(t,y) % dy1/dt = f1 = -0.5 y1 % dy2/dt = f2 = 4 - 0.1*y1 - 0.3*y2 % let y(1) = y1, y(2) = y2 % tspan = [0 1] % initial conditions y0 = [4, 6] f1 = -0.5*y(1); f2 = 4 - 0.1*y(1) - 0.3*y(2); f = [f1, f2]'; Euler Method for a System of ODEs >> [t,y]=Euler_sys('example5',[0 1],[4 6],0.5); t y1 y2 y3 ... 0.000 4.0000000000 6.0000000000 0.500 3.0000000000 6.9000000000 1.000 2.2500000000 7.7150000000 >> [t,y]=Euler_sys('example5',[0 1],[4 6],0.2); t y1 y2 y3 ... 0.000 4.0000000000 6.0000000000 0.200 3.6000000000 6.3600000000 0.400 3.2400000000 6.7064000000 0.600 2.9160000000 7.0392160000 0.800 2.6244000000 7.3585430400 1.000 2.3619600000 7.6645424576 (h = 0.5) (h = 0.2) Euler Method for Second-Order ODE function f = pendulum(t,y) % nonlinear pendulum d^2y/dt^2 + 0.3dy/dt = -sin(y)d^2y/dt^2 + 0....
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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Lecture40 - System of ODEs ❚ A system of simultaneous...

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