Maths problem

Question1

Suppose two populations evolve according to the following system of differential equations.
What is the equilibrium population size of x if a = 85, b = 162?

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Question 2

Consider two populations x and y defined by the system of differential equations,

The path starts from the point (19.6,0). Determine the value of y when t = 0.39, giving your answer to 3 decimal places.

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Question 3  

A system of differential equations,

has a particular integral given by,

When p = -41, determine the value of b (an integer).

Note
: You are not asked to find the general solution.

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Question 4

A two population model is satisfied by a system of differential equations given by,
V(x,y) = (ax,by)T
The general solution of the system of differential equations has x = αeat for some constant α.
The parameter t can be eliminated and we can express y in terms of x in the form xy = C where C is some constant.
When a = 5.31, state the value of b that will satisfy xy = C. Give your answer to 2 decimal place.

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Question 5

Matrix A has eigenvalues 5 and 2 corresponding to eigenvectors (1, 1)T and (-2, 1)T. Study the problem below. Take a = 53, b = 7*53. What is the value of k (only integer allowed)?

A=

Has a solution
=

Suppose the following system

+bt

Where a and b are constants as shown, has a general solution:

Where α,β,, C, D and K are some numbers

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Question 6

Consider the following system of differential equations. Find the value of x(0.7), given the initial conditions, x(0) = 43 and y(0) = 13. Give your answer (if non-zero) to 3 decimal places.

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Question 7

The following system of differential equations has a repeated eigenvalue 2.

=2x+3y

=2y

Suppose we choose its corresponding eigenvector to be (-1, 0)T. The initial conditions are x(0) = 5.6 and y(0) = 1.5.
Find the value of x(0.23), giving your answer to 3 decimal places.

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Question 8

Consider two populations x and y defined by the system of differential equations as follow, with k = 3.53.
Determine the value of y when t = 0.5 for the path that starts from the point (34.2, 0)? Give your answer correct to 3 decimal places.
Note: The input to the sin or cos function is measured in radians, so you must change to radian mode when using your calculator.

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Question 9

Consider the following system of differential equations. Find the value of y(1.76) given the initial conditions, x(0) = 37.3 and y(0) = 8. Give your answer (if non-zero) to 3 decimal places.

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Question 10

Suppose two populations x and y are modelled by the system of differential equations shown below where c = -18.11 and k = 1.47.
Determine the equilibrium value of x. Give your answer correct to 3 decimal places.

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