APM2611 Assignment 1 Solutions 2026

APM2611 Assignment 1 Solutions 2026
ADDENDUM B: Assignments
ASSIGNMENT 01
Due date: Wednesday, 6 May 2026
ONLY FOR YEAR MODULE
FIRST AND SECOND ORDER DIFFERENTIAL EQUATIONS,
MODELING WITH
FIRST-ORDER DIFFERENTIAL EQUATIONS
Note that all the questions will be marked therefore, it is highly
recommended to attempt all of them.

, QUESTION 1
1.1 Solve:

𝑦 ′′ − 4𝑦 ′ + 4𝑦 = 0, 𝑦(0) = 3, 𝑦 ′ (0) = 1


Solve homogeneous equation

Characteristic equation:

𝑟 2 − 4𝑟 + 4 = 0
(𝑟 − 2)2 = 0 ⇒ 𝑟 = 2 (repeated root)


General solution

𝑦(𝑥) = (𝐶1 + 𝐶2 𝑥)𝑒 2𝑥


First derivative

𝑦 ′ = 𝐶2 𝑒 2𝑥 + (𝐶1 + 𝐶2 𝑥)(2𝑒 2𝑥 )
𝑦 ′ = (𝐶2 + 2𝐶1 + 2𝐶2 𝑥)𝑒 2𝑥


Apply initial conditions

At 𝑥 = 0:

𝑦(0) = 𝐶1 = 3


Now 𝑦 ′ (0):

𝑦 ′ (0) = (𝐶2 + 2𝐶1 )𝑒 0 = 𝐶2 + 2(3) = 𝐶2 + 6
𝐶2 + 6 = 1 ⇒ 𝐶2 = −5


Final answer:

𝑦 = (3 − 5𝑥)𝑒 2𝑥