MHB Differential equations Romeo and Juliet

[martha1]

Hi all! I need to give a presentation about a problem in class, but I can't seem to figure it out. This is the problem:

Consider the system dr/dt = -j, dj/dt = r , where r (t) represents Romeo’s love (positive values) or hate (negative
values) for Juliet at time t, and j(t) similarly represents Juliet’s feelings toward Romeo

The first question I was able to answer: (a) Juliet “loves to be loved,” while Romeo is intrigued by rejection.

However, at the second question I am kinda lost:

(b) Romeo’s and Juliet’s families are enemies. This can be expressed in the initial condition (r, j) = (.,.) at time t = 0.

Because it is also an online exercise we need to do, I know both values need to be either -1,0 or 1. I also assume this has to do with the the solution having the form of a sinus wave, but I am not sure what the reasoning behind this. Also, I assume that one of the values is -1, as they are enemies and -1 represents hate, but I am not sure if this is the case for Juliet or Romeo and what then the other value should be.

Then, they continue with this question:

(c) What happens in the long run?
- They are mutually in love a quarter of the time
-They are mutually in love half the time.
- They are sometimes in love with the other but
never at the same time.
- They end up being in love.
- They end up not being in love.
- None of the above.

Which can probably also be explained with a visual representation (in sinus form) of the solution, but again, I don't understand what the reasoning behind it being a sinus wave is and also how exactly the solution would look when graphed. As I need to explain this problem in class, it would be really great if someone could explain it to me.

Any help is welcome, thanks!

 

[HOI]

Hi all! I need to give a presentation about a problem in class, but I can't seem to figure it out. This is the problem:

Consider the system dr/dt = -j, dj/dt = r , where r (t) represents Romeo’s love (positive values) or hate (negative
values) for Juliet at time t, and j(t) similarly represents Juliet’s feelings toward Romeo

The first question I was able to answer: (a) Juliet “loves to be loved,” while Romeo is intrigued by rejection.

You are not going to say what the "first question" was?

However, at the second question I am kinda lost:

(b) Romeo’s and Juliet’s families are enemies. This can be expressed in the initial condition (r, j) = (.,.) at time t = 0.

Because it is also an online exercise we need to do, I know both values need to be either -1,0 or 1. I also assume this has to do with the the solution having the form of a sinus wave, but I am not sure what the reasoning behind this. Also, I assume that one of the values is -1, as they are enemies and -1 represents hate, but I am not sure if this is the case for Juliet or Romeo and what then the other value should be.

Romeo and Juliet are members of their respective families so since their families are enemies so are they, initially, r= -1, j= -1.

Then, they continue with this question:

(c) What happens in the long run?

You have the set of first order differential equations: dr/dt= -j, dj/dt= r. There are many ways to solve that but I would differentiate the first equation again to get d^2r/dt^2= -dj/dt= -r. That gives r= A cos(t)+ B sin(t). Then j= -dr/dt= -(-A sin(t)+ B cos(t)= A sin(t)- B cos(t). The initial values are j(0)= -B= -1 and r(0)= A= -1. The solutions are r(t)= - cos(t)- sin(t), j(t)= cos(t)- sin(t).

- They are mutually in love a quarter of the time
-They are mutually in love half the time.
- They are sometimes in love with the other but
never at the same time.
- They end up being in love.
- They end up not being in love.
- None of the above.

For what values of t are both r(t)= -cos(t)- sin(t) and j(t)= cos(t)- sin(t) positive.

Which can probably also be explained with a visual representation (in sinus form) of the solution, but again, I don't understand what the reasoning behind it being a sinus wave is and also how exactly the solution would look when graphed. As I need to explain this problem in class, it would be really great if someone could explain it to me.

Any help is welcome, thanks!

 

[martha1]

Thank you! This was really helpful!
Btw, the first question was to simply fill in the names Juliet and Romeo in the correct space, that's why I had put them in bold ;)

 

[martha1]

Romeo and Juliet are members of their respective families so since their families are enemies so are they, initially, r= -1, j= -1.

One more question if you don't mind, if I wasn't given the values -1,0 and 1 to choose from (which in the original assignment I wasn't) how could I have known that the initial conditions were (-1,-1) and not, say, (-1.2,-1.2)?
Because if I plot my solutions, they can take on values lower than -1 (the minimum is -1.414).

Thanks!

 

[HOI]

"Initial conditions" are independent of the differential equation. If you are not "given" the initial conditions, there is no way you can "manufacture" them from the equation. What was the statement of the problem in the original statement?