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Tuesday, April 04, 2006 Amusing little maths tutorial question.. 4. Romeo loves Juliet, and his love for her tends to increase when she loves him; naturally, it tends to decrease if she doesn’t. We can represent this by dR/dt = aJ, where R represents Romeo’s love for Juliet (negative love means hatred, of course), where J is Juliet’s love for Romeo, and a is a positive constant. Juliet, alas, loves not wisely, nor too well: she is extremely fickle, in the sense that her love for Romeo only increases when the latter actually dislikes her, but her opinion of him goes down when he loves her. So for Juliet we have dJ/dt = −bR, where b is another positive constant. Find a second order ordinary differential equation for R, and solve it under the assumption that Romeo’s love for Juliet is at its maximum, L, when he first meets her (at t = 0), as is customary with love at first sight. Use this solution to find J(t) and graph both functions on the same diagram. Show that Romeo and Juliet are both in love only one quarter of the time. What is the meaning of the constants a and b? Find Juliet’s maximum love for Romeo. In view of the meaning of a and b, does this answer make sense? Find an expression for the time it takes for the relationship to complete one entire cycle.
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