Differential equations using phase plane analysis. Task 1 (Romeo and Juliet). Le

Question

Differential equations using phase plane analysis.

Task 1 (Romeo and Juliet). Let R and J be variables which measure the love/hate that Romeo and Juliet, respectively, have for the other star-erassed lower. We consider their relationship as described by a system of the form The parameters a, b,c, d determine the evalution of their relationship ower time. The signs of the variables a, b determine what kind of lover Romeo is, Of course, the signs of the variables c, d play exactly the same roles for Juliet. For each arrangement of passible signs of a,b explain how Romeo's feelings for Juliet change depending on the current state of things. Then make up a descriptive name for the kind of lover Romeo is given those variables. (Remember that these apply to Juliec just as well, so if you would rather, yau may work with J, e, d.) Please remember that the sign of a number is either positive, negative, or zero Explore the different kinds of things that happens when you pair poten- tial Romeos and Juliets. Which kinds of pairs are doomed to terrible fates? Which pairs make for happy coll What interesting things ples? you say?

Explanation / Answer

(1) dR/dT = aR+bJ = f(R,J) form here T is time

dR/dT = (1)(aR+bJ) dR/(aR+bJ) = dT {let us take J value as constant though it is given as a variable consider nly this equation}

integrate on both sides we get 1/a ln(aR+bJ) + k = T

similarly in second equation if we take R as constant we get 1/d ln(cR+dJ) + k' = T

we get aR+bJ = e^(a(T-k)) that is an exponential form

it comes closer to 0 only when a value is large and negative J value also similarly comes as exponential form e^(d(T-K'))

a(+ve) , b(+ve) love increases ; a(+ve), b(-ve) love increases so much with time

a(-ve) , b(+ve) love increases (since in equation -ve on both sides so get cancelled)

a(-ve) , b(-ve) love depends on b and J numerical values same thing applies to equation dJ/dT

(2) above we can see that in three cases love only increases with time only in 4th case it depends on juliet when he is totally negative for her similarly for juliet only in one case it depends on romeo

so we can conclude that love only increases with time and the couple in which love exists at least with one will remain happy

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