Page 1Math 217Exam 1Name:ID:Section:This exam has 14 questions:• 13 multiple choice worth 6 points each.• 1 hand graded worth 22 points.Important:• No graphing calculators!• For the multiple choice questions, mark your answer on the answer card.• Show all your work for the written problems. You will be graded on the ease of reading yoursolution.• You are allowed a 3 × 5 note card for the exam.1. Solve the initial value problemdydx=1 − x2y2y(0) = 1What is y(2)? Choose the closest answer.(a) −3(b) −2(c) −1(d) 0(e) 1(f) 2(g) 3(h) 4(i) 5(j) 6Page 2Math 217Exam 12. Pick the differential equation that corresponds to the given slope field.-2 -1 0 1-101(a)dydx= x2(b)dydx= y2(c)dydx= x + y2(d)dydx= y + x2(e)dydx= x(f)dydx= y(g)dydx=1x(h)dydx=1yPage 3Math 217Exam 13. Consider the differential equations below. Determine which are guaranteed a unique solutionnear the initial value according to theorems discussed in class and the textbook.I. y0= xy2/3y(0) = 0II. y0= xy2/3y(0) = 1III. y0= xy2/3y(1) = 0IV. y0= xy2/3y(1) = 1(a) Only I has a unique solution(b) Only II has a unique solution(c) Only III has a unique solution(d) Only IV has a unique solution(e) Only I and II have unique solutions(f) Only I and III have unique solutions(g) Only I and IV have unique solutions(h) Only II and III have unique solutions(i) Only II and IV have unique solutions(j) Only III and IV have unique solutions(k) None of the above.4. Consider the differential equations below. Determine which are guaranteed a unique solutionnear the initial value according to theorems discussed in class and the textbook.I. y0+1xy = ex3y(0) = 0II. y0+1xy = ex3y(0) = 1III. y0+1xy = ex3y(1) = 0IV. y0+1xy = ex3y(1) = 1(a) Only I has a unique solution(b) Only II has a unique solution(c) Only III has a unique solution(d) Only IV has a unique solution(e) Only I and II have unique solutions(f) Only I and III have unique solutions(g) Only I and IV have unique solutions(h) Only II and III have unique solutions(i) Only II and IV have unique solutions(j) Only III and IV have unique solutions(k) None of the above.Page 4Math 217Exam 15. Solve the initial value problem1xdydx−2yx2= x cos x y(π/2) = π2What is y(2)? Choose the closest answer.(a) 0(b) 3(c) 4(d) 10(e) 12(f) 15(g) 18(h) 21(i) 28(j) 30Page 5Math 217Exam 16. Solve the initial value problemdxdt+ tx2+xt= 0 x(1) = 2What is x(2)? Choose the closest answer.(a) 0(b) 1/15(c) 2/15(d) 1/5(e) 4/15(f) 1/3(g) 2/5(h) 7/15(i) 8/15(j) 3/5Page 6Math 217Exam 17. Solve the initial value problemxy2dydx= x3+ y3y(1) = 1What is y(2)? Choose the closest answer.(a) −2.1(b) −1.4(c) 0(d) 2.1(e) 2.9(f) 4.5(g) 6.8(h) 9.9(i) 11.6(j) 13.2Page 7Math 217Exam 18. Solve the initial value problem3y00− 7y0+ 2y = 0 y(0) = 5, y0(0) = 5What is y(1)? Choose the closest answer.(a) 1(b) 2(c) 5(d) 12(e) 14(f) 19(g) 26(h) 72(i) 94(j) 115Page 8Math 217Exam 19. Suppose that a ball is thrown upward from the ground with an initial velocity of 10m/s andexperiences a deceleration of 0.1v m/s2due to air resistance, where v is the velocity. Assuminga constant gravitational acceleration of −9.8m/s2, what is the maximum height that the ballreaches?Choose the closest answer.(a) 0.97m(b) 2.40m(c) 3.71m(d) 4.78m(e) 5.20m(f) 6.33m(g) 90.28m(h) 102.8m(i) 980mPage 9Math 217Exam 110. Which of the equations below is exact?I. 2x cos(y + x2) + (x + cos(y + x2)) y0= 0II. y cos(xy) dx + (x cos(xy) − 2y) dy = 0III.2xex2y + 1−ex2(y + 1)2y0= 0IV. 2x2yex2+ xex2y0= 0(a) Only I is exact(b) Only II is exact(c) Only III is exact(d) Only IV is exact(e) Only I and II are exact(f) Only I and III are exact(g) Only I and IV are exact(h) Only II and III are exact(i) Only II and IV are exact(j) Only III and IV are exact11. Consider the initial value problemdydx= 2xy + exy(0) = 1Approximate y(1) using Euler’s method with step size h = 0.5Choose the closest answer(a) 2.4(b) 2.6(c) 2.8(d) 3.0(e) 3.2(f) 3.4(g) 3.6(h) 3.8(i) 4.0(j) 4.2(k) 4.4Page 10Math 217Exam 112. Compute the WronskianW = W (sin t, t sin t)Find W (1). Choose the closest answer.(a) 0.1(b) 0.2(c) 0.3(d) 0.4(e) 0.5(f) 0.6(g) 0.7(h) 0.8(i) 0.9(j) 1.013. A population of rabbits, R(t), in thousands, satisfies the differential equationdRdt=12.53(R − 2.3)2(R − 4.1)(R − 5.6) R(0) = 5Determine limt→∞R(t).(a) −∞(b) 0(c) 2, 300(d) 2, 530(e) 4, 100(f) 5, 000(g) 5, 600(h) ∞WRITTEN PROBLEM—SHOW YOUR WORKMath 217Exam 1Name:ID:Section:Note: You will be graded on the readability of your work. Use the back of this sheet, if necessary.14. Consider the autonomous equation x0(t) =−2x4+ 4x3+ 6x2100(a) Find all the equilibrium solutions.(b) Sketch the slope field corresponding to this equation. Include several possible solutioncurves.(c) For each solution that you found in part (a), state whether it is stable, unstable, orsemistable.(d) Suppose that x(0) = 2. Approximate x(1) using Runge-Kutta with step size of
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