Question

Please help me with a differential equations question:

let R(t) represent Romeo's affection for Juliet at time t, and let J(t) represent Juliet's affection to Romeo at time t. Positive R and J love, negative represent hate. Juliet becomes more attracted to Romeo when he doesn't like her, and she becomes more repulsed by him when he likes her. Romeo, on the other hand, becomes more attracted to Juliet when she is attracted to him. Their equations can be represented as: R'=aJ, J'= -bR, a and b are positive integers. Determine the behavior of R and J over time. Will they find happiness?

Answer 1

Positive R and J represent love

Answer 2

@shubhamsrg

Answer 3

@DLS

Answer 4

@ParthKohli

Answer 5

@DLS

Answer 6

@shubhamsrg

Answer 7

@lshi1993

Answer 8

yes - thank you all! any ideas? :)

Answer 9

i guess i need to draw slope field

Answer 10

@DLS

Answer 11

2?? i didn't get it... could you explain? thanks!

Answer 12

NO.

Answer 13

\[\LARGE R'(t)=aJ\]
\[\LARGE J'(t)=bJ+(c-dR)\]
I think you can do it now

Answer 14

@DLS could you explain where does the second equation come from? and also, what's the meaning of "determine the behavior of R and J over time?" thank you.

Answer 15

@shubhamsrg @ParthKohli can do that

Answer 16

I don't know differential equations. :-|

Answer 17

Maybe
@agent0smith @terenzreignz oh and @yrelhan4 @hartnn

Answer 18

@satellite73 could you help?

Answer 19

A lot more confusing than it should be, it would seem
Let \[\large D_t\] mean the derivative with respect to time...
So..
\[\large D_t R(t) =aJ(t)\]\[\large D_t J(t)=-bR(t)\]
Leading to getting the second derivative of both sides of the first equation...
\[\large D_t^2R(t) = aD_tJ(t) = -abR(t)\]
\[\large D_t^2R(t) +abR(t)=0\]\[\large (D_t^2+ab)R(t)=0\]
and the parth should be relatively straightforward from here...

Answer 20

Guys? A peer review would be nice...

Answer 21

I'm not sure you can take the R(t) out of the expression as you did in the last line.

Answer 22

It's a method, involving homogenous diff. equations. Getting the auxilliary equation and all...

Answer 23

I don't know that much about differential equations but:
\[R'(t) = aJ(t) \]\[J(t) = \frac{ 1 }{ a }R'(t)\]\[J'(t) = \frac{ 1 }{ a }R''(t)\]
Substituting in the second equation for J'(t):
\[-bR(t) = \frac{ 1 }{ a }R''(t)\]\[R''(t) = -abR(t)\]
From there, we can see that:
If R(t) > 0, then R''(t) < 0, meaning that R(t) will eventually decrease and become negative,
If R(t) < 0, then R''(t) > 0, meaning that R(t) will eventually increase and become positive,
If R(t) = 0, then R''(t) = 0, which doesn't really say much unless both R(t) and J(t) start at 0, in which case R(t) and J(t) will equal zero for all values of t.

Which means that R(t) continuously fluctuates from positive to negative, never staying in a positive state indefinitely, so Romeo and Juliet will never love eachother indefinitely.

Answer 24

thank you so much @terenzreignz and @Fruitbasket. these are really helpful. i guess i am more favor of the substitution method çoz it makes more sense to me... :) thank you all again.