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Final Exam Review and Answers

Final Exam Review and Answers - MATH 3321 Final Exam Sample...

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MATH 3321 Final Exam — Sample Questions 1. y 2 = Cx 3 - 3 is the general solution of a differential equation. Find the equation. Answer y = 3 y 2 + 9 2 xy 2. Find the general solution of x y + 3 y = cos 2 x x 2 . Answer y = sin 2 x 2 x 3 + C x 3 . 3. Find the general solution of 2 y = y 2 + 3 4 y + xy . Answer y 2 = C (4 + x ) - 3 4. Find the general solution of y + 6 y + 9 y = 0. Answer y = C 1 e - 3 x + C 2 xe - 3 x . 5. Determine a fundamental set of solutions of y + 4 y + 13 y = 0. Answer { e - 2 x cos 3 x, e - 2 x sin 3 x } 6. The function y = - 2 e - 3 x sin 2 x is a solution of a second order, linear, homogeneous differ- ential equation with constant coefficients. What is the equation? Answer y + 6 y + 13 y = 0 7. The function y = - 2 e 3 x + 4 xe 3 x is a solution of a second order, linear, homogeneous differential equation with constant coefficients. What is the equation? Answer y - 6 y + 9 y = 0 8. Given the initial-value problem y - 7 y + 12 y = 0; y (0) = 3 , y (0) = 0. (a) Find the general solution of the differential equation. (b) Find the solution that satisfies the initial conditions. Answer (a) y = C 1 e 3 x + C 2 e 4 x . (b) y = 12 e 3 x - 9 e 4 x . 9. Given the one-parameter family y 3 = Cx 2 + 4. (a) Find the differential equation for the family. (b) Find the differential equation for the family of orthogonal trajectories. 1
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(c) Find the family of orthogonal trajectories. Answer (a) y = 2 y 3 - 8 3 xy 2 . (b) y = - 3 xy 2 2 y 3 - 8 . (c) 3 x 2 y + 2 y 3 + Cy + 16 = 0. 10. A certain radioactive material is decaying at a rate proportional to the amount present. If a sample of 100 grams of the material was present initially and after 2 hours the sample lost 20% of its mass, find: (a) An expression for the mass A ( t ) of the material remaining at any time t .
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