Table of Contents
Math 445 Lab 11: Differential Equations
Problem 1: dy/dt = sin(y+t)
Show a quiver plot in the (t,y) plane of the differential equation dy/dt = sin(y+t) over the region 0 ≤ t ≤ 2pi and -pi ≤ y ≤ pi. Use Matlab's ode45 function to compute solution curves y(t) for each of the following initial conditions, and superimposed the solution curves on the quiver plot in the indicated colors y(0) = -1 in red, y(0) = 0 in blue , y(0) = 1 in black. Be sure to set the axes tight around the quiver plot, set the unit length of the axes to be equal, label the axes, and title the graph.
Problem 2: van der Pol oscillator
(a) The van der Pol oscillator is defined by the second-order differential equation
This second-order equation can be converted to a system of two first-order equations by letting y be a 2-d vector
![$y = [y_1, y_2] = [x, ~dx/dt]$](/dokuwiki/lib/exe/fetch.php?media=wiki:latex:/img0bdd47bb42fdda5a9213bcf71efc6a82.png "Math")
.
Then
![$dy/dt = [dy_1/dt, ~dy_2/dt ] = [y_2, ~\mu (1-y_1^2) y_2 - y_1]$](/dokuwiki/lib/exe/fetch.php?media=wiki:latex:/imgc28f72e164331627853f57b6da0bd926.png "Math")
.
Use this definition of the vector 
to produce a quiver plot of the van der Pol oscillator
in the 
plane for 
, over the range 
and
. Using Matlab's ode45, compute solution curves 
for the five different initial values
![$y(0) = [0.1~ 0], [0.2~0], [1~0], [2~0],$](/dokuwiki/lib/exe/fetch.php?media=wiki:latex:/img63f3ca9faebcca9ba8d17e3c4a981476.png "Math")
and ![$[3~0]$](/dokuwiki/lib/exe/fetch.php?media=wiki:latex:/img275e9e22d65809aa2a9019430d5f9670.png "Math")
, and superimpose them on the quiver plot in red.
Hint: put the call to ode45 inside a for loop that sets the value of 
to the five different initial values.
(b) Put your work for part (a) in a script file, and run the script for a range of values of 
between -1 and 1. How does the character of the solutions of the equation change as passes through 0? Turn in plots for a negative, zero, and a positive value of , whichever values you think best illustrate the change as passes through zero.
Hint: To avoid the need to change the value of in both the script and the function file that defines 
, define as an anonymous function within your script file, so that it inherits the of
the script file.
