APM3711 ASSIGNMENT 01 SOLUTIONS 2026

UNIVERSITY OF SOUTH AFRICA
DEPARTMENT OF MATHEMATICAL SCIENCES
UNISA
MODULE: APM3711
Numerical Methods II
TUTORIAL LETTER 101/3/2026
APM3711 ASSIGNMENT 01 SOLUTIONS 2026
Due date: 30 April 2026

, QUESTION 1

Given:

𝑦 ′ = −2𝑥𝑦, 𝑦(0) = 1, ℎ = 0.25


Exact solution:
2
𝑦(𝑥) = 𝑒 −𝑥


(a) Predictor and Corrector formulas

Modified Euler (Heun predictor–corrector):

Predictor:
(0)
𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑓(𝑥𝑛 , 𝑦𝑛 )


Corrector (iterative):

(𝑘 +1) ℎ (𝑘)
𝑦𝑛+1 = 𝑦𝑛 + [𝑓(𝑥𝑛 , 𝑦𝑛 ) + 𝑓(𝑥𝑛+1 , 𝑦𝑛+1 )]
2


(b) Compute 𝒚(𝟎. 𝟐𝟓)and 𝒚(𝟎. 𝟓)

Step 1: From 𝒙𝟎 = 𝟎to 𝒙𝟏 = 𝟎. 𝟐𝟓

𝑓(𝑥, 𝑦) = −2𝑥𝑦


Predictor:

𝑓(0,1) = 0
(0)
𝑦1 = 1 + 0.25(0) = 1


Corrector iteration 1:

𝑓(0.25,1) = −2(0.25)(1) = −0.5
(1) 0.25
𝑦1 = 1 + (0 + (−0.5)) = 1 − 0.0625 = 0.9375
2


Corrector iteration 2:

𝑓(0.25,0.9375) = −2(0.25)(0.9375) = −0.46875