MATH 251Exam ISep 29, 2009Name:Student Number:Instructor:Section:There are 6 multiple choice questions and 5 partial credit questions. In order to obtainfull credit for the partial credit problems, all work must be shown. Credit willnot be given for an answer not supported by work on a partial credit problem.THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINA-TION.For multiple choice problems, write the letter of your choice in the spaceprovided below.Your Answer : Points awarded1. (5 pts)Q. 7 (10 pts)2. (5 pts) Q. 8 (10 pts)3. (5 pts) Q. 9 (15 pts)4. (5 pts) Q. 10 (20 pts)5. (5 pts) Q. 11 (15 pts)6. (5 pts)MATH 251 -Exam I-1. (5 points) Which of the following differential equations is Linear?(a) yy′+ ty = t + 1(b) ln(t2)y′′+ t3/2y′+ et+3= 0(c) ty′+ cos y = 1(d) 2y′− y = y22. (5 points) Which function is a solution of the differential equation (y′)2− 5ty = 5t2+ 1 ?(a) y(t) = t2(b) y(t) = e5t(c) y(t) = −t(d) y(t) = −15t3. (5 points) Which pair of functions form a fundamental set of solutions for the equationy′′+ 2y′+ y = 0 ?(a) y1(t) = e−t, y2(t) = te−t.(b) y1(t) = e−t, y2(t) = et.(c) y1(t) = e−t, y2(t) = e1−t.(d) y1(t) = et, y2(t) = tet.Page 2 of 9MATH 251 -Exam I-4. (5 points) Suppose that a sum of 1000 dollars is invested at an annual rate of 2%compounded continuously, and additional dep osits occur continuously at a constant rate500 dollars per year. Which one of the f ollowing initial value problems models the sumS(t) a ccumulated at any time t?(a)dSdt= 0.02S + 1000, S( 0) = 500.(b)dSdt= 0.02S + 500, S(0 ) = 1000.(c)dSdt+ 0.02S = 500, S(0) = 1000.(d)dSdt+ 0.02S = 1000, S(0) = 500.5. (5 points) The initial value problem(9 − t2)y′+ 2ty = 3t2, y(π) = 2π,is certain to exist on one of the four following intervals. Find it.(a) −∞ < t < −3.(b) −6 < t < 3.(c) 3 < t < +∞.(d) 0 < t < 2π.Page 3 of 9MATH 251 -Exam I-−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5yf(y)Figure 1: Sketch o f f (y) as a function of y.6. (5 points) Consider the following autonomous ODEdydt= f(y),where f(y) = y(y − 1)(2 − y)(y − 3)/4. A sketch of f(y) is on Figure 1 above.Which one of the f ollowing statements about the above ODE is NOTcorrect?(a) The ODE has two stable equilibrium solutions,(b) The ODE has two unstable equilibrium solutions,(c) All solutions of this ODE approach −∞, as t → +∞,(d) All negative solutions of this ODE approach −∞, as t → +∞.Page 4 of 9MATH 251 -Exam I-7. (10 points) Solve the following equation. You may leave your answer in an implicitform.dydx=sin(x)y + exp(y), where exp(y) represents the exponential function ey.Page 5 of 9MATH 251 -Exam I-8. (10 points) Solve the following initial value problem: ty′+ (t + 1)y = 1, y(1) = 1.Page 6 of 9MATH 251 -Exam I-9. (15 points)(a) Determine whether the following equation is exact.2xydx + (x2+ y2)dy = 0.(b) Solve it.Page 7 of 9MATH 251 -Exam I-10. (20 points) A tank with a capacity of 100 liters orig inally contains 2 liters of water with1 grams of salt in solution. Water containing a salt concentration of14gm/l flows intothe tank at a rate of 4 l/min, and the mixture is allowed to flow out of the tank at arate of 2 l/min.(a) Set up an initial value problem that models the dependence of the amount of saltin the tank on time t.(b) Solve the above initial value problem.(c) Find the smallest concentratio n of salt in the tank, before the mixture starts tooverflow.Page 8 of 9MATH 251 -Exam I-11. (15 points) Solve the initial value problemy′′+ 2y′+ 2y = 0, y( 0) = 1, y′(0) = −1.Page 9 of
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