MATH 251 MIDTERM EXAMINATION I February 20, 2002PENN STATE UNIVERSITYThere are 5 multiple choice and 7 partial credit problems in this examination. Each multiplechoice problem has 5 choices. Circle the correct answer. Each multiple choice problem isworth 5 points.The use of calculators during the examination is forbidden.Check your examination booklet carefully.There should be 12 problems on 10 pages.MATH 251 MIDTERM EXAMINATION I PAGE 21. (5 points) Which of the following is TRUE?a) y0=tyis a first order linear differential equationb) sin t y00+ (1 − t2)y0+ cos t y = 0 is a second order linear differential equationc) y00+ (y0)3+ y = 0 is a nonlinear differential equation of order 3d) y00+ y0+ y = t is a second order homogeneous differential equatione)∂y∂t+ ty = 0 is an ordinary differential equation2. (5 points) Let y(t) be the solution of the initial value problemdydx= y3− yy(0) =910.Then limt→∞y(t) is equal toa) ∞b)910c) 1d) 0e) −1MATH 251 MIDTERM EXAMINATION I PAGE 33. (5 points) Find the value for the constant b, for which given equation is exact.(exsin y + bx2y2)dx + (excos y + x3y)dy = 0a) b = 0b) b =13c) b = 3d) b =32e) b = 14. (5 points) Let y1(t) and y2(t) be two solutions of a second order, homogeneous, linear dif-ferential equation. Suppose the Wronskian W (y1(t), y2(t)) = e−t. Which of the following isFALSE?a) y1(t) and y2(t) are linearly independent functions.b) 2y1(t) − 3y2(t) is also a solution of the differential equation.c) y1(t) and y2(t) do not constitute a fundamental set of solutions.d) All solutions of the differential equation can be expressed as c1y1(t) + c2y2(t), wherec1and c2are constants.e) W (2y1(t), 3y2(t)) = 6e−tMATH 251 MIDTERM EXAMINATION I PAGE 45. (5 points) The largest interval on which the differential equation(t2− 1)y00+ sin t y0+ cos t y = 0, y(5) = 0, y0(5) = 1is certain to have a unique twice differentiable solution isa) (−∞, 5)b) (−1, 1)c) (5, ∞)d) (1, ∞)e) (−∞, −1)6. (5 points) Find the solution to the following initial value problem in explicit form.dydx= −xyy(0) = −1MATH 251 MIDTERM EXAMINATION I PAGE 57. (12 points) Find the general solution oft2dydt+ 3ty = et, t > 0MATH 251 MIDTERM EXAMINATION I PAGE 68. (13 points) Find the general solution of(2xy − 3x2)dx + (x2+ 2y)dy = 0(you may keep your solution in implicit form).MATH 251 MIDTERM EXAMINATION I PAGE 79. (12 points) Find the general solution of the following equations. Express your answer interms of real valued functions.a) y00− 4y0+ 4y = 0b) y00− 4y0+ 5y = 0MATH 251 MIDTERM EXAMINATION I PAGE 810. (8 points) Solve the initial value problemy00− 4y = 0y(0) = 4, y0(0) = 4MATH 251 MIDTERM EXAMINATION I PAGE 911. (12 points) A jar contains a sugar solution. Initially it had 2 liters of water and 15 gramsof dissolved sugar. Water containing 5 grams of sugar per liter enters the jar at a rate of 2liters/min. The well stirred mixture flows out at the same rate.How many grams of dissolved sugar is present in the jar after ln 5 minutes?MATH 251 MIDTERM EXAMINATION I PAGE 1012. (13 points) A ball of mass 2 kg is dropped from rest in a viscous liquid. As a result the ballexperiences a drag force which is twice the magnitude of its velocity. Find the distance theball travels in the first 2 seconds of its motion. Assume that g = 10
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