Question 1
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? dx
= dt x
10
− x
10
( y a
) dy
= dt
−y
+
5 y
5
( x b
)
Answer:
Question 2
Consider two populations x and y defined by the system of differential equations,
( ẋ ẏ
) = (
1 2
2 −2
) ( x y
)
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.
Answer:
Question 3
A system of differential equations, ẋ = −3x + 6y + e −4t ẏ = x + 3y + pe −4t has a particular integral given by,
( x p y p
) = ( a b
) e −4t
When p = -41, determine the value of b (an integer).
Note : You are not asked to find the general solution.
Answer:

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 = α e at 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.
Answer:
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= [
3 2
1 4
] dx
= 3x + 2y + t dt dy
= x + 4y + 7t dt
Has a solution
[ x y]
= [ t +
−2t −
4
5
7
10
]
Suppose the following system dx
= 3x + 2y + 4 e 3t + at dt dy
= x + 4y − e 3t
+bt dt
Where a and b are constants as shown, has a general solution:
[ x y] = C [
−2
1
] e αt + D [
1
1
] e βt + [
3
−2
] e γt
4
+ K ( t +
−2t −
5
7
10
)
Where
α,β, γ , C, D and K are some numbers
Answer:

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. dx
[ dt dy
] = [
−3 1
1 −3
] [ x y] dt
Answer:
Question 7
The following system of differential equations has a repeated eigenvalue 2. dx
=2x+3y dt dy
=2y dt
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.
Answer:
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.
dx
( dt dy
) = A ( x y
) where A = (
0 −k k 0
) , k > 0 dt
Answer:

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. dx
[ dt dy
] = [
−3 1
1 −3
] [ x y] dt
Answer:
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.
dx
( dt dy
) = ( x − y + c y + kx
) dt
Answer: