% In class 20-Mar-08 clear, clc, home fprintf('The date and time: %s \n', datestr(now))
The date and time: 08-Apr-2008 11:55:20
% Ordinary Differential Equations (ODE's) % see pages 703 and 706 in text book % % Solve IVPs via ODE45 % First order differential equation (FODE) % dy/dt = 3t % y(0) = 0 .... (initc) % for 0 <= t =< 10 sec .... (Tspan) % % Use: % [t, y] = ode45(@fun, [Tspan], [initc], p1, p2, ...) % % fun: dydt = pt; ..... (where p = 3) % function [dydt] = fun(t, y, P) % % % Second order differential equation % (d_2)y/(dt)_2 = 3dy/dt + 6t % y(0) = 0 .... (initc) % y'(0) = 3 .... (initc) % for 0 <= t =< 10 sec .... (Tspan) % % Use: % % function [dydt] = fun(t, y, P1, P2) % dydt = ? % Use change of variables to convert the 2nd order IVP to % two 1st order IVP's. % % (d_2)y/(dt)_2 = 3dy/dt + 6t ...(1) % let y_1 = y ...(2) % dy/dt = dy_1/dt = y_2 ...(3) % (d_2)y/(dt)_2 = dy_2/dt ...(4) % % Sub (2), (3), and (4) into (1) % % dydt = [y_2; 3*y_2 + 6*t]; % % 5th order differential equation % % Same as above but: (d_6)y/(dt)_6 % % dydt = [y_2; y_3; y_4; y_5; y_6; 3*y_2 + 6*t] % or % dydt = [y_2 % y_3 % y_4 % y_5 % y_6 % 3*y_6 + 6*t] % % % Note: % ODE45 handles "smooth" or "non-jerky" curves % ODE23 handles "jerkey" curves (ie. stock market graphs) % % Do the following excersizes in book: 12.12 % Use parameter passing with loops. % dy/dt = t^2/q - y % % Use: % Function [dydt] = fun12a(t, t, q) % dydt = ((t.^2)./p) - y_1 % % For solving ODE's with a constant reppresented with a letter % for j = 1 to 5 % if j == 1 % p = 2 % elseif j == 2 % p = 3 % . % . % . % % end % next % % [t, y] = ode45(@fun12a, [], [], [], p) % % The output vector y contains y and all the missing derivatives % Ex: dy/dt = 2t then y_t % Ex: (d_2)y/(dt)_2 = 2t then y_t, y'(t) ... y'(t) = dy/dt(t) % Ex: (d_3)y/(dt)_3 = 2t then y_t, y'(t), y''(t), y'''(t) %