x(t). ii) Identify the long-term behavior of the population system, i.e., compute the lim00 Note how this limit is related to one of the eigenvectors related to the system matrix A. iii) Suppose that the initial conditions a, b are non-negative and a + b > 0. We want the population of vehicles to be distributed according to the ratio 1:2 as t →∞. What should the values of k AB, kBA be in order to achieve this?
x(t). ii) Identify the long-term behavior of the population system, i.e., compute the lim00 Note how this limit is related to one of the eigenvectors related to the system matrix A. iii) Suppose that the initial conditions a, b are non-negative and a + b > 0. We want the population of vehicles to be distributed according to the ratio 1:2 as t →∞. What should the values of k AB, kBA be in order to achieve this?
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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![ii) Identify the long-term behavior of the population system, i.e., compute the lim+00 x(t).
Note how this limit is related to one of the eigenvectors related to the system matrix A.
iii) Suppose that the initial conditions a, b are non-negative and a + b > 0. We want the
population of vehicles to be distributed according to the ratio 1: 2 as t → xo. What should
the values of kAB, kBA be in order to achieve this?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2Fffa4b6b2-fe0d-4bd7-a402-068a94939587%2Fghcdi5c_processed.png&w=3840&q=75)
Transcribed Image Text:ii) Identify the long-term behavior of the population system, i.e., compute the lim+00 x(t).
Note how this limit is related to one of the eigenvectors related to the system matrix A.
iii) Suppose that the initial conditions a, b are non-negative and a + b > 0. We want the
population of vehicles to be distributed according to the ratio 1: 2 as t → xo. What should
the values of kAB, kBA be in order to achieve this?
![Problem 4. Suppose that you have two spatial locations A and B. You have a population
of self-driving vehicles, that switch from location A to B, with probability KABAT,and from
location B to A with probability kBAAt. This can be represented by a "reversible chemical
reaction", with "reaction rates" kAB and and kBA
A KAB, B
kBA
В
The population of self-driving vehicles in each location A and B is given by XA(t) and
XB(t), respectively. For At arbitrarily small, given that kABand kba are non-negative con-
stant values, the evolution of the population of vehicles in site A and B are given by,
*a(t)
-KABTA(t) + KBATB(t)
(1)
KABTA(t) – KBATB(t)
(2)
TA(0) :
= a xB(0) = b
(3)
i) Find the solution to the IVP.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2Fffa4b6b2-fe0d-4bd7-a402-068a94939587%2Fyti1fik_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 4. Suppose that you have two spatial locations A and B. You have a population
of self-driving vehicles, that switch from location A to B, with probability KABAT,and from
location B to A with probability kBAAt. This can be represented by a "reversible chemical
reaction", with "reaction rates" kAB and and kBA
A KAB, B
kBA
В
The population of self-driving vehicles in each location A and B is given by XA(t) and
XB(t), respectively. For At arbitrarily small, given that kABand kba are non-negative con-
stant values, the evolution of the population of vehicles in site A and B are given by,
*a(t)
-KABTA(t) + KBATB(t)
(1)
KABTA(t) – KBATB(t)
(2)
TA(0) :
= a xB(0) = b
(3)
i) Find the solution to the IVP.
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