UNIVERSITY OF PRETORIA
DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 256 -
DIFFERENTIAL EQUATIONS
ONLINE Examination WINTER SCHOOL Presented in January 2021
22 January 2021
Total: 50
External Examiner : Dr D Moubandjo
Time: 180 minutes to do the examination Internal Examiners : Ms L Mostert
Ms M Ohlhoff
Surname & Initials: Student Nr:
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SIGNATURE:
The paper consists of 2 pages and 11 questions (as well as a list of Laplace transforms).
SHOW ALL YOUR WORK
Read the following instructions
Prohibited
1. No notes, textbook or any other help is allowed. It is a closed book examination.
2. Calculators MAY NOT be used.
Format
1. Only handwritten submissions in one pdf file will be accepted.
2. Write the declaration, your surname, initials, student number and your signature
clearly on the first page of your submission.
3. Numerate your pages and keep them in the correct order and the correct way up.
4. Label each question (sub-question) clearly.
5. Draw a line with a ruler at the end of each question.
6. Show all your work and calculations and indicate how you use definitions and/or apply
theorem
Submission
1. You have 180 minutes to do the examination.
2. You have 25 minutes to prepare your submission.
3. You have 10 minutes to submit.
4. Name your pdf when scanning your answers (to avoid submitting the wrong document).
5. Make sure that you are satisfied with your answers before submitting.
6. You have only ONE attempt to do the examination. A second attempt can only be
considered if the content is the same as the first attempt and it is clear that you had to
submit twice due to submission difficulties.
7. If you can’t submit because the link is not available any more it means that you start
submitting after the submission time lapsed. It will therefore not be accepted.
Copyright reserved
LAPLACE TRANSFORMS
L {f 0 (t)} = sF (s) − f (0)
L {f 00 (t)} = s2 F (s) − sf (0) − f 0 (0)
L {eat f (t)} = F (s − a)
L {f (t − a)U(t − a)} = e−as F (s), L {g(t)U(t − a)} = e−as L{g(t + a)}
L{tn f (t)} = (−1)n F (n) (s), L{tf (t)} = −F 0 (s)
Rt
L {f (t) ∗ g(t)} = L { 0 f (τ )g(t − τ )dτ } = F (s)G(s)
Rt F (s)
L{ 0 f (τ )dτ } = s
1
L {1} = s
n!
L {tn } = sn+1
1
L {eat } = s−a
s
L {cos kt} = s2 +k2
k
L {sin kt} = s2 +k2
s
L {cosh kt} = s2 −k2
k
L {sinh kt} = s2 −k2
e−as
L {U(t − a)} = s
Question 1
Find a general implicit solution of the homogeneous differential equation
dy y 2 3y
y − = 2x sec2 ( ).
dx x x
[4]
Question 2
Consider the initial value problem
dy 4x
−( 2 )y = 2xy −1/2 (x2 + 2)3 sin(x2 ), y(0) = 4.
dx x +2
Classify the differential equation and then determine an explicit solution of the initial
value problem.
[5]
Question 3
dy y−5
Consider the initial value problem = , y(a) = b.
dx y+2
3.1 Use a suitable theorem to determine the value(s) of b for which the initial value
problem will have at most one solution.
[1.5]
3.2 Sketch the graph of the solution of the initial value problem
dy y−5
= , y(0) = 4
dx y+2
without solving the initial value problem. The graph must show significant facts clearly.
[2]
3.3 Solve the initial value problem in 3.2 to determine the x-intercept of the graph in
question 3.2.
[3]
Question 4
Consider the differential equation y (4) − 4y (3) + 5y 00 = 3x2 − 2e2x cos x + 4xex .
4.1 Determine a general solution, yc , of the associated homogeneous differential equation.
[3]
4.2 Give only the form of a particular solution, yp , of the non-homogeneous differential
equation. Do not determine the coefficients.
[3]
Question 5
Use the method of undetermined coefficients to solve the following initial value problem
1 2
y 00 + 10y 0 + 16y = 6e−2t + 9et , y(0) = , y 0 (0) = − .
3 3
[5]
Question 6
5s − 3
6.1 Determine L−1 e−2s 2
. SHOW all steps.
4s + 8s + 13
[4]
3s2
( )
6.2 Determine L−1 ln( 2 ) .
s + 8s + 16
[3]
1
Question 7
( )
−1 1
Given that L = te3t .
(s − 3)2
( )
−1 1
7.1 Use the convolution of two relevant functions to determine L .
(s + 3)(s − 3)2
[2]
7.2 Use the Laplace-transform to solve the initial value problem
y 00 − 6y 0 + 9y = −4e−3t , y(0) = 0, y 0 (0) = 6.
[2]
Question 8
(
cos t 0 ≤ t < 3π
Let f (t) =
4 t ≥ 3π
Find L{f (t)} without using the definition of the Laplace transform.
[3]
Question 9
At time t = 0 the bottom plug of a full rectangular water tank of height 10 meter and
volume 2500m3 is removed. The tank drains in T minutes. The differential equation that
describes the flow√of the water through the opening at the bottom is given by Torricelli’s
law A(h) dh
dt
= −k h.
Give a mathematical model, that is a differential equation and initial conditions, that
describes the situation. Do not solve.
[1.5]
Question 10
Confirm that the coefficient matrix of the system
x01 = −4x1 + x2
x02 = −13x1 +2x2
x03 = 2x3
has one real eigenvalue and two complex eigenvalues. Then use the eigenvalue-eigenvector
method to determine a general real-valued solution of the system of differential equations.
Show all steps and give your answer in vector form.
[6]
Question 11
The following initial value problem describes a typical mixing problem with xi (t) the
amount of salt (in kilogram) in tank Bi at time t (in minutes).
−1 0 0 x1 (t) 27
0
X̄ = AX̄ = 1 −2 0
X̄, X̄(t) = x2 (t) , X̄(0) = 0 .
0 2 −3 x3 (t) 0
The eigenvalues and corresponding
eigenvectors of the
matrix A are
1 0 0
λ1 = −1 , K̄1 = 1 , λ2 = −2 , K̄2 = 2 , λ3 = −3 , K̄3 = 0 .
1 4 3
Give the solution of the initial value problem and then determine the amount of salt in
tank B2 at time t = ln 5. Simplify your answer as far as possible.
[2]
