Juliet is in love with Romeo, who happens (in our version of the story) to be a fickle lover. The more Juliet loves him, the more he begins to dislike her. When she hates him, his feelings for her warm up. On the other hand, her love for him grows when he loves her and withers when he hates her. A model for this ill-fated romance is
dj/dt = Ar
dr/dt = -Bj,
where A and B are positive constants, r(t) represents Romeo's love for Juliet, and j(t) represents Juliet's love for Romeo at time t. (Negative love represents hate.)
a) The constant on the right hand side of Juliet's equation (dj/dt) has a positive sign, while the constant in Romeo's equation is negative. Explain why this follows from the story.
b) Derive a second-order differential equation for r(t) and solve it. (Your equation should involve r and its derivatives, but not j and its derivatives.)
c) Express r(t) and j(t) as functions of t, given r(0)=1 and j(0)=0. Your answer will contain A and B.
d) As you may have discovered, the outcome of the relationship is a never-ending cycle of love and hate. Find what fraction of the time they both love each other.
*snickers* Come to think of it, this describes at least one (out of not very many) of my past romantic relationships fairly accurately. Now to hope for a better model for any future relationships.