MAT3700 ASSIGNMENT 2 SEM 1 2023

MAT3700
ASSIGNMENT 2
2023

, QUESTION 1




Solution:



𝑑2 𝑦 𝑑𝑦
+ 3 + 2𝑦 = 4𝑥 2
𝑑𝑥 2 𝑑𝑥


𝐴𝑢𝑥𝑖𝑙𝑖𝑎𝑟𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛: 𝑚2 + 3𝑚 + 2 = 0



𝑚2 + 3𝑚 + 2 = 0
(𝑚 + 1)(𝑚 + 2) = 0

𝑚1 = −1 𝑜𝑟 𝑚2 = −2



𝐶𝑜𝑚𝑝𝑙𝑖𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝑦𝑐 = 𝐶1 𝑒 𝑚1 𝑥 + 𝐶2 𝑒 𝑚2 𝑥



𝑦𝑐 = 𝐶1 𝑒 𝑚1 𝑥 + 𝐶2 𝑒 𝑚2 𝑥

𝑦𝑐 = 𝐶1 𝑒 −𝑥 + 𝐶2 𝑒 −2𝑥



𝐵𝑦 𝑚𝑒𝑡ℎ𝑜𝑑 𝑜𝑓 𝑢𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑑 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ∶



𝑃𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝑦𝑝 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶



𝑦𝑝 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶

𝑦 ′ 𝑝 = 2𝐴𝑥 + 𝐵

𝑦 ′′ 𝑝 = 2𝐴



𝑦 ′′ 𝑝 + 3𝑦 ′ 𝑝 + 2𝑦𝑝 = 4𝑥 2

2𝐴 + 3(2𝐴𝑥 + 𝐵) + 2(𝐴𝑥 2 + 𝐵𝑥 + 𝐶) = 4𝑥 2

, 2𝐴 + 6𝐴𝑥 + 3𝐵 + 2𝐴𝑥 2 + 2𝐵𝑥 + 2𝐶 = 4𝑥 2

2𝐴𝑥 2 + (6𝐴 + 2𝐵)𝑥 + (2𝐴 + 3𝐵 + 2𝐶) = 4𝑥 2



∴ 2𝐴 = 4 ⇒ 𝐴=2



∴ 6𝐴 + 2𝐵 = 0
6(2) + 2𝐵 = 0
12 + 2𝐵 = 0
2𝐵 = −12
𝐵 = −6



∴ 2𝐴 + 3𝐵 + 2𝐶 = 0
2(2) + 3(−6) + 2𝐶 = 0
4 − 18 + 2𝐶 = 0
−14 + 2𝐶 = 0
2𝐶 = 14
𝐶=7



𝑦𝑝 = 𝐴𝑥 2 + 𝐵𝑥 + 𝐶

𝑦𝑝 = 2𝑥 2 − 6𝑥 + 7



𝑦(𝑥) = 𝑦𝑐 + 𝑦𝑝

𝑦(𝑥) = 𝐶1 𝑒 −𝑥 + 𝐶2 𝑒 −2𝑥 + 2𝑥 2 − 6𝑥 + 7



𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:

𝑦(𝑥) = 𝐶1 𝑒 −𝑥 + 𝐶2 𝑒 −2𝑥 + 2𝑥 2 − 6𝑥 + 7