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MAT3700 Assignment 02 Answers (Year Module) 2024
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This paper contains Mathematics III (Engineering ) Assignment 02 Solutions with clear step by step calculations. Distinction Guaranteed.
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Mathematics III (Engineering)Assignment 2 Answers
Semester 1 2024
0027 65 934 4052
,
,
,QUESTION 1
𝑑2𝑖 𝑑𝑖
+ 2 + 2𝑖 = 85 sin 3𝑡 … … … … … (1)
𝑑𝑡 2 𝑑𝑡
𝑚2 + 2𝑚 + 2 = 0
−2 ± √(2)2 − (4)(1)(2)
𝑚=
2(1)
−2 ± √−4
𝑚=
2
𝑚 = −1 ± 𝑗
𝑖𝐶.𝐹 = 𝑒 −𝑡 (𝐴 cos 𝑡 + 𝐵𝑠𝑖𝑛 𝑡)
𝑖𝑝 = 𝐶 sin 3𝑡 + 𝐷 cos 3𝑡
𝑖𝑝′ = 3 cos 3𝑡 − 3𝐷 sin 3𝑡
𝑖𝑝′′ = −9𝐶 sin 3𝑡 − 9𝐷 cos 3𝑡
Substituting into equation (1);
−9𝐶 sin 3𝑡 − 9𝐷 cos 3𝑡 + 2(3𝐶 cos 3𝑡 − 3𝐷 sin 3𝑡) + 2(𝐶 sin 3𝑡 + 𝐷 cos 3𝑡) = 85 sin 3𝑡
−9𝐶 sin 3𝑡 − 9𝐷 cos 3𝑡 + 6𝐶 cos 3𝑡 − 6𝐷 sin 3𝑡 + 2𝐶 sin 3𝑡 + 2𝐷 cos 3𝑡 = 85 sin 3𝑡
−7𝐶 sin 3𝑡 − 7𝐷 cos 3𝑡 + 6𝐶 cos 3𝑡 − 6𝐷 sin 3𝑡 = 85 sin 3𝑡
Comparing terms in sin 3𝑡
−7𝐶 − 6𝐷 = 85 … … … … … . (3)
−7𝐷 + 6𝐶 = 0 … … … … … . . (4)
, Solving equations (3) and (4) simultaneously;
(3) × 7: − 49𝐶 − 42𝐷 = 595 … … … … . (5)
(4) × 6: 36𝐶 − 42𝐷 = 0 … … … … . . … (6)
(5) − (6) : − 85𝐶 = 595
𝐶 = −7
From equation (3)
−7(−7) − 6𝐷 = 85
49 − 6𝐷 = 85
𝐷 = −6
𝑖𝑝 = −7 sin 3𝑡 − 6 cos 3𝑡
𝒊𝒈𝒆𝒏 = 𝒆−𝒕 (𝑨 𝐜𝐨𝐬 𝒕 + 𝑩 𝐬𝐢𝐧 𝒕) − 𝟕 𝐬𝐢𝐧 𝟑𝒕 − 𝟔 𝐜𝐨𝐬 𝟑𝒕
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