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apm255finalexam2

# apm255finalexam2 - APM255 Fall 2007 Final Exam Instructor W...

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1 APM255 Fall 2007 Final Exam Instructor: W. Zhang Student’s Name: Mark: Instruction: Close books and notes. Show your work and justify your answers. Calculators permitted. 1. (10 pts.) Identify if the following ordinary differential equations (ODEs) to be linear or nonlinear, separable (or non-separable), and autonomous or non-autonomous. . (a) 0 cos 3 ln = + x yx x y y e x (b) 0 1 2 / 1 = + y y (c) x y y y cos ) 1 ( 4 3 = 2. (10 pts.) A tank initially contains 50 gal of pure water. Brine containing 2 lb of salt per gallon enters the tank at the rate of 5 gal/min., and the well-mixed solution leaves the tank at the rate of 3 gal/min. Set-up (do not solve) the ODE initial value problem that describe the amount of salt, x(t) , the tank contains at time t.

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2 3. (30 pts.) Solve the general and singular solutions (if there are) of the following ODEs. (a) x y y y cos ) 1 ( 4 3 =
3 (b) 0 3 3 = + x y y x

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4 4. (15 pts.) Show by direct computation that ) )( )( ( 1 1 1 det 2 2 2 a c c b b a c b a c b a = .
5 5. (15 pts.) Find a basis and the dimension of the solution space of

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