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Quiz-3-f10_Sol - AMS 361 Applied Calculus IV by Prof Y Deng...

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1 AMS 361: Applied Calculus IV by Prof. Y. Deng Quiz 3 Tuesday (11/16/2010) at 5:20-6:40PM Open Book and Open Own Lecture Notes but no Electronics (1) Do Any Two of the Three Problems; Mark the Two Problems You Attempt; Will Only Grade the First Two if You Don’t Mark the Problems You Attempted; (2) Each Problem Is Worth 5 Points; (3) No Points for Guessing Work and for Solutions without Appropriate Intermediate Steps; Partial Credits are given only for Steps that are Relevant to the Solutions. SB ID: Class ID: Name: Problems To Grade? Score Remarks Q2-1 Q2-2 Q2-3 Total Score Q3-1 (5 Points): Find the general solution of the following non-homogeneous DE using any of your favorite method: 𝑥 2 𝑦 ′′ + 2 𝑥𝑦′ − 6 𝑦 = 72 𝑥 5 𝑥 = 𝑒 𝑡 , 𝑡 = ln( 𝑥 ) 𝑥 𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑡 𝑥 2 𝑦 ′′ = 𝑑 2 𝑦 𝑑𝑡 2 𝑑𝑦 𝑑𝑡 𝑦 ′′ − 𝑦 + 2 𝑦 6 𝑦 = 72 𝑒 5𝑡 𝑦 ′′ + 𝑦 6 𝑦 = 72 𝑒 5𝑡 Characteristic Solution 𝑟 2 + 𝑟 − 6 = 0 ( 𝑟 + 3)( 𝑟 − 2) = 0 𝑟 1 = 3, 𝑟 2 = 2 𝑦 𝑐 = 𝑐 1 𝑒 −3𝑡 + 𝑐 2 𝑒 2𝑡
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2 Particular Solution 𝑦 𝑝 = 𝐴𝑒 5𝑡 𝑦 𝑝 = 5 𝐴𝑒 5𝑡 𝑦 𝑝 ′′ = 25 𝐴𝑒 5𝑡 𝑦 𝑝
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