UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology
⋄
ASSIGNMENT 01
Semester 1 – 2026
⋄
Module Code: APM3711
Module Name: Numerical Methods II
Assignment No.: 01
Due Date: 30 April 2026
Semester: Semester 1, 2026
Submitted in partial fulfilment of the requirements for Numerical Methods II
at the University of South Africa.
, UNISA | APM3711 Numerical Methods II – Assignment 01
Question 1: Modified Euler (Predictor–Corrector) Method [14 marks]
Question: Consider the initial value problem
y ′ = −2xy, y(0) = 1.
2
The exact solution is y(x) = e−x . Use the Modified Euler (predictor–corrector) method with
step size h = 0.25 to approximate the solution up to x = 0.5. At each step perform two
correction iterations.
1(a) Predictor and Corrector Formulae
Question: Write down the predictor formula and the corrector formula used in the Modified
Euler method.
Let f (x, y) = −2xy.
Predictor (Euler step):
(0)
yn+1 = yn + h f (xn , yn )
Corrector (iterated): For k = 1, 2, . . .
(k) hh
(k−1)
i
yn+1 = yn + f (xn , yn ) + f xn+1 , yn+1
2
1(b) Numerical Computation of y(0.25) and y(0.5)
Question: Compute the approximate numerical values y(0.25) and y(0.5) using the Modified
Euler method with two correction iterations at each step.
Given: h = 0.25, f (x, y) = −2xy, x0 = 0, y0 = 1.
Step 1: From x0 = 0 to x1 = 0.25
Compute f (x0 , y0 ):
f (0, 1) = −2(0)(1) = 0
Page 1 of 11
College of Science, Engineering and Technology
⋄
ASSIGNMENT 01
Semester 1 – 2026
⋄
Module Code: APM3711
Module Name: Numerical Methods II
Assignment No.: 01
Due Date: 30 April 2026
Semester: Semester 1, 2026
Submitted in partial fulfilment of the requirements for Numerical Methods II
at the University of South Africa.
, UNISA | APM3711 Numerical Methods II – Assignment 01
Question 1: Modified Euler (Predictor–Corrector) Method [14 marks]
Question: Consider the initial value problem
y ′ = −2xy, y(0) = 1.
2
The exact solution is y(x) = e−x . Use the Modified Euler (predictor–corrector) method with
step size h = 0.25 to approximate the solution up to x = 0.5. At each step perform two
correction iterations.
1(a) Predictor and Corrector Formulae
Question: Write down the predictor formula and the corrector formula used in the Modified
Euler method.
Let f (x, y) = −2xy.
Predictor (Euler step):
(0)
yn+1 = yn + h f (xn , yn )
Corrector (iterated): For k = 1, 2, . . .
(k) hh
(k−1)
i
yn+1 = yn + f (xn , yn ) + f xn+1 , yn+1
2
1(b) Numerical Computation of y(0.25) and y(0.5)
Question: Compute the approximate numerical values y(0.25) and y(0.5) using the Modified
Euler method with two correction iterations at each step.
Given: h = 0.25, f (x, y) = −2xy, x0 = 0, y0 = 1.
Step 1: From x0 = 0 to x1 = 0.25
Compute f (x0 , y0 ):
f (0, 1) = −2(0)(1) = 0
Page 1 of 11
