2.3 Guess and Test

October 5, 1998

An example with real exponential solutions

Consider the equation

d^2*y/dt^2+2*dy/dt-3*y = 0 .

It has exponential solutions y = e^st for numbers s satisfying the associated quadratic equation

s^2+2*s-3 = 0 ,

that is, s = -3 and s = 1 .  Below is a phase portrait of the associated system

dy/dt = v ,

dv/dt = 3*y-2*v .

> with(DEtools):

> phaseportrait([D(y)(t)=v(t),D(v)(t)=3*y(t)-2*v(t)],
[y(t),v(t)],t=0..0.7,
[[y(0)=1,v(0)=-3],[y(0)=1,v(0)=1],[y(0)=-1,v(0)=3],[y(0)=-1,v(0)=-1]]);

[Plot]

Note that these particular solution curves are straight lines. They should be, v = sy , after all. Clearly the other solution curves are more complicated.

An example without real exponential solutions

Consider the equation

d^2*y/dt^2+2*dy/dt+50*y = 0 .

It has exponential solutions y = e^st for numbers s satisfying the associated quadratic equation

s^2+2*s+50 = 0 ,

that is, s = -1+7*i and s = -1-7*i . Below is a phase portrait of the associated system

dy/dt = v ,

dv/dt = -50*y-2*v .

> phaseportrait([D(y)(t)=v(t),D(v)(t)=-50*y(t)-2*v(t)],
[y(t),v(t)],t=0..2,
[[y(0)=1,v(0)=0],[y(0)=-1,v(0)=0]],
linecolor=[navy,green],
stepsize=0.01);

[Plot]

> plot([exp(-t)*sin(7*t),diff(exp(-t)*sin(7*t),t)],t=0..4,
title="y in red, v in green");

[Plot]