APM2611
ASSIGNMENT 1 2024
UNIQUE NO.
DUE DATE: 8 MAY 2024
, APM2611/101/0/2024
ADDENDUM A: ASSIGNMENTS
ASSIGNMENT 01
Due date: Wednesday, 8 May 2024
-
ONLY FOR YEAR MODULE
First order separable, linear, Bernoulli, exact and homogeneous equations. Higher order
homogeneous DE’s. Solving non-homogeneous DE’s using the undetermined
coefficients, variation of parameters and operator methods.
Answer all the questions. Show all your own and personalized workings, you get ZERO
to a question if we see that you have copied someone’s else solution word by word.
If you choose to submit via myUnisa, note that only PDF files will be accepted.
Note that all the questions will be marked therefore, it is highly recommended to attempt all of them.
Question 1
Prove that the given function is a solution of the given differentialequation.
1.
y00− 2y 0 + 5y = 0 ; y = e x (cos 2x − sin 2x)
2.
dy π
− ecos x = −y cot x , y( ) = 4;
dx 2
y = − csc x(e cos x − 5)
3. (i) Verify that y = θ 1 (x) = x 2 and y = θ 2 (x) = −x 2 are solutions of the differential equation
xy 0 − 2y = 0 on the interval (−∞, ∞).
(ii) Verify that the piecewise-direction function
−x 2 , x<0
y= x 2, x≥0
is a solution of xy 0 − 2y = 0 on the interval (−∞, ∞).
Question 2
Consider the DE
(6xy − y 3)dx + (4y + 3x 2 − 3xy 2 )dy = 0.
1. Prove that it is exact.
2. Solve it.
11
, Question 3
Solve the following DEs:
1.
dy
x2 + 2xy = 5y 3
dx
2.
x 2 y0 = y 2
+ 2y + 1, y(1) = 13
Question 4
1. Solve the given differentialequations by separation of variables:
(i)
dy
ex y −y
= e + e−2x−y
dx
(ii)
2
dx y+1
y ln |x| =
dy x
2. Solve the initial value problem:
dy y2 − 1
= 2 , y(2) = 2
dx x −1
3. Solve the following DEs using the appropriate substitutions.
(a)
dy y
x = y + xe x , y(1) = 1
dx
(b)
xy − y 2
y0 = Hint: make substitution y = ux
x2 − y 2
Leave your answer in an implicit form.
(c)
x 2 y00+ 2xy 0 + y = 0
Question 5
1. A large tank is filled with 500 litres of pure water. Brine containing 2kg of salt per litre is
pumped into the tank at a rate of 5 litres/min. The well-mixed solution is pumped out at the
same rate. Find the number A(t) of kg of salt in the tank at time t. What is the concentration
of the solution in the tank at t = 5min?
2. Solve part (1.) under the assumption that the solution is pumped out at a faster rate of 10
litres/min. When is the tank empty?
– End of assignment –
12
ASSIGNMENT 1 2024
UNIQUE NO.
DUE DATE: 8 MAY 2024
, APM2611/101/0/2024
ADDENDUM A: ASSIGNMENTS
ASSIGNMENT 01
Due date: Wednesday, 8 May 2024
-
ONLY FOR YEAR MODULE
First order separable, linear, Bernoulli, exact and homogeneous equations. Higher order
homogeneous DE’s. Solving non-homogeneous DE’s using the undetermined
coefficients, variation of parameters and operator methods.
Answer all the questions. Show all your own and personalized workings, you get ZERO
to a question if we see that you have copied someone’s else solution word by word.
If you choose to submit via myUnisa, note that only PDF files will be accepted.
Note that all the questions will be marked therefore, it is highly recommended to attempt all of them.
Question 1
Prove that the given function is a solution of the given differentialequation.
1.
y00− 2y 0 + 5y = 0 ; y = e x (cos 2x − sin 2x)
2.
dy π
− ecos x = −y cot x , y( ) = 4;
dx 2
y = − csc x(e cos x − 5)
3. (i) Verify that y = θ 1 (x) = x 2 and y = θ 2 (x) = −x 2 are solutions of the differential equation
xy 0 − 2y = 0 on the interval (−∞, ∞).
(ii) Verify that the piecewise-direction function
−x 2 , x<0
y= x 2, x≥0
is a solution of xy 0 − 2y = 0 on the interval (−∞, ∞).
Question 2
Consider the DE
(6xy − y 3)dx + (4y + 3x 2 − 3xy 2 )dy = 0.
1. Prove that it is exact.
2. Solve it.
11
, Question 3
Solve the following DEs:
1.
dy
x2 + 2xy = 5y 3
dx
2.
x 2 y0 = y 2
+ 2y + 1, y(1) = 13
Question 4
1. Solve the given differentialequations by separation of variables:
(i)
dy
ex y −y
= e + e−2x−y
dx
(ii)
2
dx y+1
y ln |x| =
dy x
2. Solve the initial value problem:
dy y2 − 1
= 2 , y(2) = 2
dx x −1
3. Solve the following DEs using the appropriate substitutions.
(a)
dy y
x = y + xe x , y(1) = 1
dx
(b)
xy − y 2
y0 = Hint: make substitution y = ux
x2 − y 2
Leave your answer in an implicit form.
(c)
x 2 y00+ 2xy 0 + y = 0
Question 5
1. A large tank is filled with 500 litres of pure water. Brine containing 2kg of salt per litre is
pumped into the tank at a rate of 5 litres/min. The well-mixed solution is pumped out at the
same rate. Find the number A(t) of kg of salt in the tank at time t. What is the concentration
of the solution in the tank at t = 5min?
2. Solve part (1.) under the assumption that the solution is pumped out at a faster rate of 10
litres/min. When is the tank empty?
– End of assignment –
12
