3f-fall2010-final_exam_sample

# 3f-fall2010-final_exam_sample - Math 3F Sample Final Exam...

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Unformatted text preview: Math 3F Sample Final Exam Laney College, Fall 2010 Fred Bourgoin 1. Solve each differential equation. (a) dy dx = x- y x (b) ( x + 1) dy dx =- y + 10 2. A tank initially contains 100 gal of brine in which there is dissolved 20 lb of salt. Starting at time t = 0, brine containing 3 lb of dissolved salt per gallon ﬂows into the tank at the rate of 4 gal/min. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously ﬂows out of the tank at the same rate. (a) How much salt is in the tank at the end of 10 minutes? (b) When is there 160 lb of salt in the tank? 3. Compute the Laplace transform of f ( t ) = te 4 t using the definition. 4. Solve the initial-value problem. y ′′ + 6 y ′ + 13 y = 0 , y (0) = 3 , y ′ (0) =- 1 5. Solve the differential equation. y ′′- 10 y ′ + 25 y = 30 x + 3 6. A spring is such that a force of 4 N would stretch it 5 cm. A 2-kg mass is attached at the end of the spring. The spring is pulled 2 cm beyond its natural length and released at t = 0 with velocity 4cm/sec. The damping constant is γ = 16. (a) Find the displacement of the mass at time t . (b) Is the system overdamped, underdamped, or critically damped? 7. Use Laplace transforms to solve the initial-value problem. y ′′ + 5 y ′ + 4 y = 0 , y (0) = 1 , y ′ (0) = 0 1 8. Solve the linear system x ′ = ( 1- 5 8- 12 ) x + ( 3 e 2 t- e 2 t ) . 9. Solve the linear system { x ′ = 6 x- y y ′ = 5 x + 2 y . 2 Math 3F Sample Final Exam Solutions Laney College, Fall 2010 Fred Bourgoin 1. Solve each differential equation....
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3f-fall2010-final_exam_sample - Math 3F Sample Final Exam...

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