APM2611 Assignment 03 2025 (Accurate Solutions) Due 13 August 2025.

APM2611

ASSIGNMENT 03

Due 13 August 2025

, Assignment 03 Solutions

Question 1

Problem Statement

Solve the following initial value problem using Laplace transforms:


y ′′ + 7y ′ + 10y = 9 cos t + 7 sin t, (1)


with initial conditions y(0) = 5, y ′ (0) = 4.


Step 1: Take the Laplace Transform

Apply the Laplace transform to both sides of the differential equation:


L{y ′′ + 7y ′ + 10y} = L{9 cos t + 7 sin t}.


L{y ′′ (t)} = s2 Y (s) − sy(0) − y ′ (0) = s2 Y (s) − 5s − 4,

L{y ′ (t)} = sY (s) − y(0) = sY (s) − 5,

L{y(t)} = Y (s),
s 1 9s + 7
L{9 cos t + 7 sin t} = 9 · +7· 2 = 2 .
s2 +1 s +1 s +1
Thus,
9s + 7
(s2 Y (s) − 5s − 4) + 7(sY (s) − 5) + 10Y (s) = .
s2 + 1

Step 2: Simplify the Left-Hand Side

Combine terms:
9s + 7
(s2 + 7s + 10)Y (s) − 5s − 4 − 35 = .
s2 + 1
9s + 7
(s2 + 7s + 10)Y (s) − 5s − 39 = .
s2 + 1
9s + 7 14s + 46
(s2 + 7s + 10)Y (s) = + 5s + 39 = .
s2 + 1 s2 + 1

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