APM3706 Assignment 02 Solutions 2026

APM3706 Assignment 02 Solutions 2026
UNISA

,ASSIGNMENT 02
STUDY GUIDE: CHAPTERS 2 and 3
Answer All Questions
In All assignments we may mark all or few of those questions.
Question 1
(i) Solve Exercise 2.4 (b) from SG.
(ii) Solve Exercise 2.32 (c) from SG.
(iii) Solve Exercise 3.16 from SG.

Question 2.1

Given
𝟒 −𝟏 𝟏
𝑿̇ = [ ] 𝑿, 𝑿(𝟎) = [ ]
𝟗 𝟒 𝟏


Let
𝟒 −𝟏
𝑨=[ ]
𝟗 𝟒


We solve using eigenvalues and eigenvectors.



Find the eigenvalues

The characteristic equation is

∣ 𝑨 − 𝝀𝑰 ∣= 𝟎
𝟒−𝝀 −𝟏
∣ ∣= 𝟎
𝟗 𝟒−𝝀
(𝟒 − 𝝀)𝟐 + 𝟗 = 𝟎
(𝟒 − 𝝀)𝟐 = −𝟗
𝟒 − 𝝀 = ±𝟑𝒊


Hence

𝝀𝟏 = 𝟒 + 𝟑𝒊, 𝝀𝟐 = 𝟒 − 𝟑𝒊




Find an eigenvector for 𝝀 = 𝟒 + 𝟑𝒊

, Solve

(𝑨 − 𝝀𝑰)𝒗 = 𝟎
−𝟑𝒊 −𝟏 𝒙
[ ][ ] = 𝟎
𝟗 −𝟑𝒊 𝒚


From the first row:

−𝟑𝒊𝒙 − 𝒚 = 𝟎
𝒚 = −𝟑𝒊𝒙


Choose 𝒙 = 𝟏,
𝟏
𝒗=[ ]
−𝟑𝒊


Write

𝒗 = 𝒑 + 𝒊𝒒


where
𝟏 𝟎
𝒑 = [ ],𝒒 = [ ]
𝟎 −𝟑




Form the real solution

For eigenvalue

𝝀 = 𝜶 + 𝜷𝒊 = 𝟒 + 𝟑𝒊


the real solutions are

𝑿𝟏 = 𝒆𝟒𝒕 (𝒑𝐜𝐨𝐬⁡ 𝟑𝒕 − 𝒒𝐬𝐢𝐧⁡ 𝟑𝒕)
𝑿𝟐 = 𝒆𝟒𝒕 (𝒑𝐬𝐢𝐧⁡ 𝟑𝒕 + 𝒒𝐜𝐨𝐬⁡ 𝟑𝒕)


Therefore
𝐜𝐨𝐬⁡ 𝟑𝒕
𝑿𝟏 = 𝒆𝟒𝒕 [ ]
𝟑𝐬𝐢𝐧⁡ 𝟑𝒕


and