Shakespearean Differential Equations

Date: 10/27/98 at 06:47:34
From: Saxby Brown
Subject: Advanced Integral Calculus Question

I was wondering if you could help me out with a hard (and somewhat 
quirky) calculus question. It has to do with real world rates:

Having broken up with Romeo, Juliet falls in love with Geraldo. He 
responds to her affections, but as his feelings grow he is increasingly 
frightened by the prospect of commitment. Indeed dG/dt = aJ - bG (where 
G denotes Geraldo's love for Juliet, J denotes Juliet's love for 
Geraldo, a is a positive constant representing Geraldo's 
responsiveness, and b is another positive constant representing his 
fear of commitment. Juliet's feelings change at a rate proportional to 
Geraldo's love for her, according to the differential equation 
dJ/dt = cG where c is a constant (not necessarily positive).

(a) Show that, if d^2 < -4ac, the relationship oscillates between love 
    and hate, with the intensity of the emotions decreasing with time.
(b) Now suppose that d^2 > -4ac > 0. Solve the equations above and thus 
    determine the fate of the relationship. Does this depend on the 
    initial conditions?
(c) Now find the general solutions J(t) and G(t) in the case c > 0. 
    Show that the fate of the relationship depends on the sign of the 
    quantity 2cG(init)+(b+sqr(b^2+4ac))J(init), where G(init) and 
    J(init) are the initial values of G and J.

If you have any hints, it would be greatly appreciated. Thanks in 
advance. You guys are a really great help sometimes!

Saxby Brown

Date: 10/27/98 at 09:58:22
From: Doctor Rob
Subject: Re: Advanced Integral Calculus Question

Differentiate the first equation with respect to t. Substitute for 
dJ/dt its equal from the second equation. That will give you a 
homogeneous second-order differential equation with constant 
coefficients for G. You can do a similar thing to get an equation for 
J, and it turns out that they are the same equation, 
F" + b*F' - a*c*F = 0. 

Now you have to solve this equation. The solutions look like:

   F(t) = A*e^(r*t) + B*e^(s*t)

where r, s, A, and B are constants, r and s are the two roots of the
quadratic equation x^2 + b*x - a*c = 0, and A and B are constants
determined by the initial conditions.

The sign of b^2 + 4*a*c tells you whether r and s are real or complex.

That should be enough hints. You finish.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/