# The Famous Romeo and Juliet Problem

## Main Question or Discussion Point

I've been trying to solve the Romeo and Juliet problem in differential equations:

Romeo is in love with Juliet, but in our version of the story, Juliet
is a fickle lover. The more Romeo loves her, the more Juliet wants
to run away and hide. But when Romeo gets discouraged and
backs off, Juliet begins to find him strangely attractive. Romeo,
on the other hand, tends to echo her: he warms up when she loves
him, and grows cold when she hates him.

So let R = Romeo's love for Juliet and
let J = Juliet's love for Romeo

Then
$$\frac{dR}{dt}=aJ, \frac{dJ}{dt}=-bR$$

How might I solve this pair of differential equations? Is it possible? I have only taken calculus I and II by the way so my knowledge of DE is limited.

Thanks!

NOTE: This isn't for homework, just from a leisure book I'm reading. Some of you might know it!

BiP

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Easy, just differentiate then plug in.

Then it's second order homogenous, which are easy to solve.

Is that Strogatz btw?

HallsofIvy
Homework Helper
Another way to do this is to treat it as a single "vector" problem. Let
$$V(t)= \begin{pmatrix}R(t) \\ J(t)\end{pmatrix}$$
then
$$\frac{dV}{dt}= \begin{pmatrix}\frac{dR}{dt} \\ \frac{dR}{dt}\end{pmatrix}= \begin{pmatrix}aJ(t) \\ bR(t)\end{pmatrix}$$
or
$$\frac{dV}{dt}= \begin{pmatrix}0 & a \\ b & 0\end{pmatrix}\begin{pmatrix}R(t) \\ J(t)\end{pmatrix}$$

That matrix has two distinct eigenvalues and so can be diagonalized.

You can also easily write the solution with the help of the matrix exponential. But I guess that is not what you want.

I always wonder when people start to apply differential equations as model for some situations, do they really understand why the differential equation describes the stated situation. In your example, do you understand why the equations are as they are to describe your Romeo/Julia love story?

You can also easily write the solution with the help of the matrix exponential. But I guess that is not what you want.

I always wonder when people start to apply differential equations as model for some situations, do they really understand why the differential equation describes the stated situation. In your example, do you understand why the equations are as they are to describe your Romeo/Julia love story?
Yep! I just have trouble solving it.

BiP

I would recommend to google for "linear differential equations" and look for similar examples.
impossible to find the solution in this case.

Alternatively, also the Laplace transfom (which I actually prefer) could be used
to calculate the solution.

My second standard question is always:

Could you solve

$$x'(t) = -k \cdot x(t) , \quad x(0)=x^0$$

Another way to do this is to treat it as a single "vector" problem. Let
$$V(t)= \begin{pmatrix}R(t) \\ J(t)\end{pmatrix}$$
then
$$\frac{dV}{dt}= \begin{pmatrix}\frac{dR}{dt} \\ \frac{dR}{dt}\end{pmatrix}= \begin{pmatrix}aJ(t) \\ bR(t)\end{pmatrix}$$
or
$$\frac{dV}{dt}= \begin{pmatrix}0 & a \\ b & 0\end{pmatrix}\begin{pmatrix}R(t) \\ J(t)\end{pmatrix}$$

That matrix has two distinct eigenvalues and so can be diagonalized.
Whoa there. He said he's taken up to calc 2, and you're talking eigenvalues and diagonalization to solve differential equations? There's no way he's going to understand that with just calc 2 knowledge.

Whoa there. He said he's taken up to calc 2, and you're talking eigenvalues and diagonalization to solve differential equations? There's no way he's going to understand that with just calc 2 knowledge.
Ya that's what I said, but then the physics department at my university was all "meh meh meh, that's just how the sequence goes--take diffeqs and linear algebra next semester, after you've already taken waves and vibrations."

Struggling super hard.