Love calculator

Romeo loves Juliet, but Juliet likes to play hard to get and finds Romeo's excessive enthusiasm rather repulsive- the more he loves her, the less she likess him. On the other hand, when he loses interest, she fears losing him and begins to see his good side. Romeo is more straightforward: His love for Juliet increases when she is warm to him, and decreases when she is cold. Let R(t) represent Romeo's feelings and J(t) Juliet's. 

The lovers feelings are given:
{ dR/dt = aJ   ,     R(0) = 10000
{ dJ/dt = -bR  ,     J(0) = 0

where a, b are positive constants and m,n, represent their feelings when they first meet.
(Evidently, Romeo loves Juliet deep deep at first sight, and Juliet, of course, is indifferent)

dR/dt = aJ ------- (1)
dJ/dt = -bR ------ (2)

therefore, d/dt (1) =>  d2R/dt2 = a dJ/dt ------- (3)

substitute (2) into (3) => d2R/dt2 +abR = 0       (2nd order homo differential equation)

charactreristic equation:
 L^2 +ab = 0
 L = +- sqrt(ab) i        (i.e. a complex number)

solving:
 R = C1 cos sqrt(ab) t  +  C2 cos sqrt(ab) t
 J = -C1 sqrt(b/a) sin sqrt(ab) t  +  C2 sqrt(b/a) cos sqrt(ab) t

putting in initial conditions, C1 =  10000, C2 = 0.

R = 10000 cos sqrt(ab) t
J = -10000 sqrt(b/a) sin sqrt(ab) t

Over 1 period of time, they are only in love a quater of the time in 3pi/2sqrt(ab) < t < 2pi/sqrt(ab) i.e. both positive. They are out of love three quaters of the time....


Moral of the story:

Girls: honey, please DO NOT play hard to get.
Guys: have some faith macho, shes lagging due to her period.