APM3706 Assignment 1 solutions 2026

Ordinary Differential Equations
APM3706 Assignment 1 solutions 2026
Year module
Department of Mathematical Sciences
This tutorial letter contains
Assignment 01.
APM3706/201/0/2026
ASSIGNMENT 01
STUDY GUIDE: CHAPTERS 1
Answer All Questions
In All assignments we may mark all or few of those questions

, QUESTION 1

1.1 Check if system is degenerate
Given:
𝑥̇ + 𝑦̇ + 𝑦 = 𝑒 𝑡 (1)
𝑥̈ + 𝑦̈ + 𝑦̇ = 𝑒 𝑡 (2)

Differentiate (1)
𝑑
(𝑥̇ + 𝑦̇ + 𝑦) = 𝑥̈ + 𝑦̈ + 𝑦̇
𝑑𝑡

So:
𝑥̈ + 𝑦̈ + 𝑦̇ = 𝑒 𝑡

This is exactly equation (2).
Conclusion:
• Equation (2) is dependent on (1)
• Therefore, the system is degenerate
Solution type
Since equations are consistent:
• Infinitely many solutions
General solution:
Let:
𝑥̇ + 𝑦̇ + 𝑦 = 𝑒 𝑡

Choose 𝑦freely, then solve for 𝑥:
𝑥̇ = 𝑒 𝑡 − 𝑦̇ − 𝑦

Integrate:
𝑥 = ∫ (𝑒 𝑡 − 𝑦̇ − 𝑦)𝑑𝑡
𝑥 = 𝑒 𝑡 − 𝑦 − ∫ 𝑦 𝑑𝑡 + 𝐶

1.2 Solve system (eliminate y)
(𝐷 + 5)𝑥 + (𝐷 + 1)𝑦 = 𝑒 𝑡 (1)
(𝐷 + 11)𝑥 + (2𝐷 + 1)𝑦 = 6(2)

Eliminate y
Multiply (1) by 2𝐷 + 1:
(2𝐷 + 1)(𝐷 + 5)𝑥 + (2𝐷 + 1)(𝐷 + 1)𝑦

Multiply (2) by 𝐷 + 1:
(𝐷 + 1)(𝐷 + 11)𝑥 + (𝐷 + 1)(2𝐷 + 1)𝑦