How to solve ODEs with imaginary eigenvalues?

On Tue, 8 Apr 2003, Stavros Macrakis wrote:

> I am no expert on differential equations, but...

> -- Doesn't the eigenvector technique only work for *linear* differential
> equations?  The I^3 term is not linear....

Nor am I.  The problem is for a class I'm taking.  (The text is "Mathematical
Modeling" by Meerschaert - problem 6-16)  Part of the problem is to take the
nonlinear system given by the equations

    L: 1$
    K: 1/5;
    
    f1 : (-I^3 - 4 * I - V) / L$
    f2 : I / K$

and get the corresponding linear system by evaluation at V = I = 0.  That's
what the lines

    Amat: matrix ([diff (f1, I), diff (f1, V)], [diff (f2, I), diff (f2, V)])$
    Bmat: Amat, I:0, V:0$

should do, I think.  Then I call

    evals: eigenvalues (Bmat);

to get the eigenvalues of the linear system, and the solution to that should
be:

    soln: A * exp (t * (evals [1][1])) + B * exp (t * (evals [1][2]));
 
                                                      {   ?t      -?t}
This is ok for K > 1/4, where the solutions are real  {A E   + B E   },
where the ? are some real numbers.  But for K < 1/4, I get complex eigenvalues.
Applying demoivre (and I wonder where that name comes from?) gives

        -2t                           -2t
    B %E   (%I SIN(t) + COS(t)) + A %E   (COS(t) - %I SIN(t))
                                      
so I think from here (that DE course was a LONG time ago!) I take the
real & imaginary parts, to get the general solution.  But I don't know
how to get Maxima to do that, and indeed am not real sure how I would do
it by hand: that DE course was a LONG time ago!  Any hints would be welcome.

This example seems pretty simple: maybe it could be cleaned up a bit and
the nonlinear term removed, and used in the documentation as a "how to
solve system of linear ODEs" example?

> -- What do you mean "Maxima uses C for its own purposes"?  Are you
> thinking of the %C that Maxima uses for introducing "arbitrary
> constants"?  %C is not the same as C.

I was thinking of the Cn used as prompt strings.  Maybe just a plain C wouldn't
conflict, but why chance it?

James