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gibson:teaching:fall-2012:math445:lab11

# 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 .

Then .

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 and , 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. 