When two chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation dX/dt = k(a-X)(B-X) where k 0 is a constant of proportionality and B a 0. Here X(t) denotes the number of grams of the new compound formed in time t. (a) Describe the behavior of X as t - infinity. (b) Consider the case when a = B. What is the behavior of X as t - infinity when X(0) a. (c) Find the exact solution of the DE if k=1, a=B, and satisfying x(0)=a/2. Does the behaviour of t - infinity agree with ans (b)?

Question

When two chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation

dX/dt = k(a-X)(B-X)

where k > 0 is a constant of proportionality and B > a > 0. Here X(t) denotes the number of grams of the new compound formed in time t.

(a) Describe the behavior of X as t -> infinity.
(b) Consider the case when a = B. What is the behavior of X as t -> infinity when X(0) < a. When X(0) > a.
(c) Find the exact solution of the DE if k=1, a=B, and satisfying x(0)=a/2. Does the behaviour of t -> infinity agree with ans (b)?

aum · Answer

You can solve the DE using partial fractions and then answer the other questions.

anonymous · Answer

So must I answer part (c) first?