Math 217 Practice Exam 1. Page Which of the following differential equations is exact?

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1 Page 1 1. Which of the following differential equations is exact? (a) (3x x 3 )dx + (3x 3x 2 ) d = 0 (b) sin(x) dx + cos(x) d = 0 (c) x 2 x 2 = 0 (d) (1 + e x ) + (2 + xe x ) d = 0 CORRECT dx (e) e x dx e x d = 0 (f) (2x ) + (2x 2 + x 2 ) = 0 (g) x x2 dx + ln()d = Suppose a population of deer P (t) satisfies the initial value problem What is lim t P (t)? (a) 0 (b) 1000 (c) 2000 (d) 5000 CORRECT (e) (f) dp dt = 2(P 1000)(5000 P )2, P (0) = Suppose a water tank initiall contains a 100 gal mixture of salt and water. Brine at a concentrate of 0.5 lbs/gal flows into the tank at the same rate the well mixed brine flows out of the tank. How much salt should initiall be in the tank so that the amount of salt stas constant as the brine flows through the tank, i.e., so equilibrium is acheived? (a) 100 lbs (b) 200 lbs (c) 150 lbs (d) 250 lbs (e) 50 lbs CORRECT (f) 0 lbs

2 Page 2 4. Which of the following differential equations best represents the slope field below? x -1.0 (a) = x + CORRECT (b) = x (c) = x (d) = x + 1 (e) = x + 2 (f) = x + 1

3 Page 3 5. Consider the following differential equation Which of the following are true? dx dt = kx2 (3 x)(x 1). I. If k > 0 then x = 3 is a stable critical point. II. If k > 0 then x = 1 is a stable critical point. III. If k < 0 then x = 1 is a stable critical point. (a) I onl (b) II onl (c) III onl (d) I and II (e) I and III CORRECT (f) II and III (g) All are true (h) None are true

4 Page 4 6. Identif the slope field of the differential equation d dx = x2 + sin. - x - x - x (a) (b) (c) - x - (d) CORRECT - x - x - - (e) (f)

5 Page 5 7. Consider the table of value for the two variable function f(x, ). (x, ) f(x, ) (0, 1) 1 (1, 0) 1 (1, 1) 2 (2, 1) 5 If n is the n-th approximate using the improved Euler method with step size h = 1 of the IVP what is 3? d dx = f(x, ), (0) = 1 The word improved should not be here. Improved Euler method will not be on the test. (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 (f) 5 (g) 6 CORRECT (h) 7 (i) None of the above 8. Solve the initial value problem d dx = x 1 (0) = 3 2 What is (2)? (a) 0 (b) 1 (c) 2 (d) 3 CORRECT (e) 4 (f) 5 (g) 6 (h) None of the above

6 Page 6 9. Consider the differential equations below. Determine which are guaranteed a unique solution near the initial value according to theorems discussed in class and the textbook. I. = (x 1)( 2) 2/3 (1) = 1 II. = (x 1)( 2) 2/3 (1) = 2 III. = (x 1)( 2) 2/3 (2) = 2 IV. = (x 1)( 2) 2/3 (2) = 1 (a) Onl I has a unique solution (b) Onl II has a unique solution (c) Onl III has a unique solution (d) Onl IV has a unique solution (e) Onl I and II have unique solutions (f) Onl I and III have unique solutions (g) Onl I and IV have unique solutions CORRECT (h) Onl II and III have unique solutions (i) Onl II and IV have unique solutions (j) Onl III and IV have unique solutions (k) None of the above. 10. Solve the initial value problem d dx = (ln ln x + 1) (1) = 2 x What is (2)? (a) 0 (b) 2 (c) 4 (d) 6 (e) 8 CORRECT (f) 10 (g) None of the above

7 Page Find the general solution to (2x 3 + 4x 3 ) d dx = 3x , (a) = Cx (b) 4 + x 2 2 = Cx 3 (c) x x 2 2 = Cx 4 e x (d) 4 + x 2 4 = Cx 2 e x (e) 4 + x 2 2 = Cx 5 (f) = Cx 5 CORRECT 12. Which of the following equations has = 1 as a stable critical point? I) = 2 1, II) = 1, III) = (a) I onl (b) II onl (c) III onl CORRECT (d) I and II onl (e) I and III onl (f) II and III onl (g) I, II, and III 13. At noon a car starts from rest at point A and proceeds at constant acceleration along a straight road toward point B. If the car reaches B at 12:50 P.M. with a velocit of 60 mi/h, what is the distance from A to B? (a) 25 miles CORRECT (b) 27 miles (c) 29 miles (d) 31 miles (e) 33 miles (f) 35 miles

8 Page Solve d dx + = cos x (0) = 1. e x What is (π)? (a) 2e π (b) e π + 1 (c) e π 2 (d) e π 1 (e) + 1 e π 1 (f) CORRECT e π 15. Solve the differential equation ( cos x + 2xe ) + (sin x + x 2 e 3 2 ) d dx = 0 (a) cos x + 2xe = C (b) sin x + x 2 e 3 2 = C (c) 2 cos x + 2xe = C (d) sin x + x 2 e 3x 2 = C (e) sin x + 2xe 3 = C (f) cos x + x 2 e 3 = C (g) sin x + x 2 e 3 2 = C (h) cos x + 2xe 3 = C The correct answer is not a choice. (i) sin x + x 2 e 3 = C CORRECT

9 Page 9 Written Problem: Show our work. 16. Consider the following autonomous equation: = Solution: (a) Notice = 2 ( 1) 3 ( + 1) so the equilibria solutions are 1, 0, (b) (c) = 1 is stable, = 0 is semistable, = 1 is unstable. x

10 Page 10 Written Problem: Show our work. 17. A 100L tank has 50L of a 2 kg/l salt solution in it. Pure water is flowing in to the tank at 10L/min, and the well mixed solution is draining from the tank at 5L/min. Solution: (a) t = 10 min. (b) x(10) = 50 kg (c) x(20) = 50/e 18.4 kg