7. ODE Systems

[ Objectives | Background | Exercises ]

OBJECTIVES

To understand 2nd order ODEs as systems of first order ODEs.

Become acquainted with pplane5 and plot some solution curves for systems of ODE's.

BACKGROUND

Let us consider the second order differential equation

x''(t)=g(t,x(t),x'(t)).

By introducing the new function y(t)=x'(t) we may rewrite this equation as the coupled pair of first order equations

x'(t)=y(t)

y'(t)=g(t,x(t),y(t)).

More generally one is often interested in considering more general coupled differential equation of the form

x'(t)=f(t,x(t),y(t))

y'(t)=g(t,x(t),y(t)),

where f and g are given functions. A good way to interpret these equations is as follows: (x(t),y(t)) is the position of a leaf subject to wind whose velocity at the space-time point (x,y,t) has x-component given by f(t,x,y) and y-component given by g(t,x,y). The MATLAB program "pplane5" should help you understand this interpretation in the case where f and g do not depend on t.

EXERCISES

Start by reading about pplane5 on pages 87 -- 97 in Polking and Arnold.

Write the differential equation x''(t)+a*x'(t)+2x(t)=0 as a coupled pair of first order differential equations.

Use pplane5 to plot out some solution curves in "phase space" (i.e. in the (x,y)=(x,x') plane) to the system you found in part 1 for a = -1,0, and 1. (Hint you may use parameters in pplane5.) Write a short explanation on the correlation of the roots of the characteristic polynomial
r^2+ar+2=0
and the observed behavior of your graphs.

Use pplane5 to plot out some solution curves to the system of equations

x'=sin(y-x/5)

y'=y^2*sin(x)/4.

Make the plot as interesting as you can. You should hand in your pictures and explanation for this lab.

[ Objectives | Background | Exercises ]

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Last modified on November 16, 1999 at 11:01 AM.