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Quiz-1-s11-Solution - AMS 361 Applied Calculus IV by Prof Y...

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1 AMS 361: Applied Calculus IV by Prof. Y. Deng Quiz 1 Wednesday (03/09/2011) at 3:50PM-5:10PM in 137 Harriman (1) Closed Book with 1-page (double-sided 8.5x11) formulas. (2) Do any two of the three problems. If all three are attempted, the two best (and only two) will be graded. (3) Each problem is worth 5 points. (4) 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 Q1-1 Q1-2 Q1-3 Total Score Q1-1 (5 points): It is obvious that 𝒚 = 𝒂 satisfies the following DE. With this info, find its general solution 𝒅𝒚 𝒅𝒙 = [ 𝒇 ( 𝒙 ) + 𝒚 + 𝒂 ]( 𝒚 − 𝒂 ) where 𝒇 ( 𝒙 ) is a given well-behaved function and “ a ” is a given constant. Solution: Substition: 𝑦 = 𝑎 + 1 𝑧 We get 𝑑𝑦 𝑑𝑥 = 1 𝑧 2 𝑑𝑧 𝑑𝑥 Substitute to the original DE, we get 𝑑𝑧 𝑑𝑥 + ( 𝑓 ( 𝑥 ) + 2 𝑎 ) 𝑧 + 1 = 0 Which is a 1 st order linear DE and solvable with the integrating factor method:
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