New version of Romeo and Juliet...... maths

hey, guys and girls,

I've encountered a very funny mathematics ORDINARY DIFFERENTIAL EQUATION in my lecture notes. So, i thought that i might be able to share this with you all..... Don't bang me if you find it not funny..... hehe...

SYSTEMS OF FIRST-ORDER ODEs
ROMEO AND JULIET
We all know that many (most) relationships have their ups and downs. Let’s try to model this fact. Romeo loves Juliet, but Juliet believes in a more subtle approach and finds Romeo’s excessive enthusiasm rather repulsive - the more he loves her, the less she likes 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. We can model the lovers’ feelings by:
 dR/ dt = aJ ; R(0) = w
dJ/dt = −bR ; J(0) = x

where a, b are positive constants and and represent their feelings when they first meet. This is a system of simultaneous first order ODEs. In this case, the equations are linear, so it is easy to solve them. Put, as a trial,
R = Ae(mt) ; J = Be(mt)
where m could turn out to be complex - if so, we will as usual interpret the exponential to mean
that we are really dealing with sine and cosine functions. [The final solutions for R and J must
be real — the feelings are real, not complex!]
So Ame(mt) = aBe(mt) and Bme(mt) = −bAe(mt).
So we get Am = aB, Bm = −bA so ignoring
special cases we get m^2 = −ab < c =" R(0)" d ="R'(0)/(ab)^1/2=" e =" J(0)" f ="J˙(0)/(ab)^1/2=" x=" 0"> 0.
Then
R(t) = w.cos((ab)^1/2.t)
J(t) = −w.(b/a)^1/2.sin((ab)^1/2.t)
Then the graphs of R(t) and J(t) are: {can't paste the graph here.. basically is a sine and a cosine curves}
-Actually it is more useful to eliminate t and get a direct relation between R and J. A bit of
algebra will convince you that:
R2/R2(max)+J2/J2(max)= 1 which can be sketched in the R−J plane: it is an ellipse.
This is just for these particular initial conditions.Different initial conditions will result in
a smaller or a larger ellipse. The full set of ALL possible love-affairs is represented by an infinite
set of concentric ellipses: So this diagram tells us everything there is to know about love.
Note that at a particular time t, R(t) and J(t) have definite values, giving a point (R(t), J(t))
in these pictures. The arrows indicate the direction of motion of such a point as time goes
by. You can check this against the graphs on page 5 - notice that J becomes negative immediately after t = 0 if the initial coordinates are (R, J) = (w, 0). A picture like this, where we
have two functions R(t) and J(t) but where we eliminate t (and regard it as a parameter)
is called a PHASE PLANE DIAGRAM. It tells us many things that are not so obvious from the diagram on page 5. For example,suppose we ask: can Romeo and Juliet ever have a steady relationship with R= constant and J=constant? The answer is clear from the picture on page 7: this is possible only if R = J = 0 (at the centre). We say that this is a point of equilibrium, and clearly it is the only one. Furthermore, it is obvious from the diagram that this equilibrium is stable. (This says that Romeo and Juliet never have the typical long last happy ending unless they are both clever enough to just stay friends from the beginning .....)

Reference:
Mcinnes, Brett Chapter 7, Systems of First Order ODE, Romeo and Juliet, Dept of Mathematics, NUS.

special credit to professor for creating such a funny and educating teaching material for students studying MA1506.

** I know most of u might be having your finals or preparing for them now... so, GOOD LUCK, BFIANS....... Our friendships is like Mr.Kali's teh tarik plastic container......... ALWAYS FULL!
jia you.. gambateh... tambah minyak!!! and happy belated april fool... anyone got conned?

tikus.