Subject: How to solve ODEs with imaginary eigenvalues?
From: James Frye
Date: Mon, 7 Apr 2003 15:02:00 -0700 (PDT)
Hi,
I have a question on solving a system of ODEs, where the eigenvalues have
imaginary values. If I remember my long-ago differential equations
class, this ought to mean there's a solution which (by Euler's formula?)
can be expressed in terms of sines and cosines. So how do I get Maxima
to give me that?
Here's a concrete example. It's an RLC circuit, with variable R as a
function of I and V. So I get the equations from the problem statement as
dI/dt = (-1/L) (I**3 + 4I + V)
dV/dt = I/K
(Where I use K for capacitance instead of C, 'cause Maxima uses C for its
own purposes.)
I translate that into Maxima input, (with specified values of L & K), and solve,
thusly:
L: 1$
K: 1/5;
f1 : (-I^3 - 4 * I - V) / L$
f2 : I / K$
A: matrix ([diff (f1, I), diff (f1, V)], [diff (f2, I), diff (f2, V)])$
B: A, I:0, V:0$
evec: eigenvectors (B);
evals: eigenvalues (B);
Maxima gives me back
[[-%I - 2, %I -2], [1, 1]]
for the eigenvalues. Now where do I go from here? Is there a Maxima
command to give the solutions, or am I completely off track?
Thanks,
James