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Math 445 lecture 21; Ordinary Differential Equations

Lecture involves

  • physics ⇒ ordinary differential equation (ODE)
  • manipulating ODE until it fits form Matlab can integrate
  • Matlab's ode23, ode45 functions, plotting results
  • perhaps some idea of how Matlab integrates ODE


Pendulum physics, F = ma with linear air resistance

 m l \theta'' = -mg sin \theta - b \theta'     

 \theta'' + c \theta' + g/l \sin \theta = 0

This is a 2nd-order nonlinear differential equation in one variable, theta. 2nd order b/c $\theta''$ term. Nonlinear b/c $\sin \theta$. Linearization for small theta would approximate sin theta = theta for small $\theta$. No big thing

Get ODE into Matlab form

Matlab can integrate 1st-order nonlinear ODEs in n variables, of form

dx/dt = f(t,x)

for vector x. Fortunately we can always convert an nth-order ODE in 1 variable to a 1st order ODE in n variables, as follows. Let

  x1 = theta
  x2 = theta'


dx1/dt = d theta/dt = x2
dx2/dt = d theta'/dt = theta'' 
       = - c theta' - g/l sin theta
       = -c x2 - g/l sin x1

So for vector x = (x1, x2),

dx/dt = f(t,x) = (x2,  -c x2 - g/l sin x1)

Integrating ODE in Matlab

Let's code this as an anonymous function in Matlab, setting constants prior

c = 0;    % no air resistance
g = 9.8;  % gravity in mks (m/s^2)x
l = 1.0;  % one meter pendulum 
f = @(t,x) [x(2); -c*x(2) - g/l * sin(x(1))];
% integrate system of ODEs from t=0 to t=100, from initial condition
% [x1; x2] = [theta; theta'] = [pi/10; 0], using Matlab's ode45
[t, x] = ode45(f, [0 20], [pi/10; 0])
% outputs are an 
% N-vector t of timesteps
% N x 2 matrix x whose cols are x1, x2 = theta, theta' at the N timesteps
% e.g. x(:,1) is N-vector of theta at times corresponding to N-vector t. 
% can graph theta versus t as follows
plot(t, x(:,1), 'b')
gibson/teaching/fall-2013/math445/lecture21.txt · Last modified: 2013/12/03 05:50 by gibson