MATH 251Examination IFebruary 26, 2009Name:Student Number:Section:This exam has 15 questions for a total of 100 points. In order to obtain full credit for partialcredit problems, all work must be shown. Credit will not be given for an answer notsupported by work. The point value for each question is in parentheses to the right of thequestion number.You may not use a calculator on this exam. Please turn off and put away yourcell phone.1 :2thru11:12:13:14:15:Total:Do not write in this box.MATH 251 EXAMINATION I February 26, 20091. (9 points) Consider the list of differential equations below.A. y′= y3− y2B. y′′′− y2= sin 3tC. y′′+ ety′= 1D. t2+ y + (t + y2)y′= 0E. y′′− t3y′+ e2ty = 0F. y′′− 2y + 5y = 9 − eyG. y′+ 2y = πH. y′− t2y = t ln tFor each part, write down the letter corresponding to the equation on the list with the specifiedproperties. There is only on e correct answer to each part.(a) (3 points) First order linear equation that is not separable.(b) (3 points) Exact equation that is not separable.(c) (3 points) Second order homogeneous linear equation.Page 2 of 11MATH 251 EXAMINATION I February 26, 20092. (5 points) Consider the initial value problemsin(t) y′′+ tan(t) y′+ t y = et, y(π4) =4π3, y′(π4) = −π4.Without solving the equation, what is the largest interval in which a unique solution is guar-anteed to exist?(a) (−∞, ∞)(b) (−π2,π2)(c) (0,π2)(d) (π2,3π2)3. (5 points) Which pair of functions below cannotbe a fundamental set of solutions?(a) 4, 2 + 3t(b) cos 5t, −2 sin 5t(c) e−t, 4te−t(d) 2e−5t, −6e−5tPage 3 of 11MATH 251 EXAMINATION I February 26, 20094. (5 points) Find the solution of the initial value problemy′=e4ty, y(0) = −2.(a) y = −r12e4t+72(b) y =12e4t−52(c) y =r12e4t−92(d) y = −√2e4t5. (5 points) Find the general solution of the exact equation3x2y3− yexy+ (3x3y2− xexy+ 2) y′= 0.(a) x3y3− exy+ 2y = C(b) 2x3y3− exy= C(c) 9x2y2− exy− xyexy= C(d)yexy− 3x2y33x3y2− xexy+ 2= CPage 4 of 11MATH 251 EXAMINATION I February 26, 20096. (5 points) Suppose a mass-spring system described by the equationu′′+ k u = 2 cos 4t − sin 4tis undergoing resonance. What is the value of the spring constant k?(a) 0(b) 2(c) 4(d) 167. (5 points) A 100-liter vat initially contains 80 liters of 2 grams/liter sodium hydroxide solution.At t = 0, sodium hydroxide solution with a concentration of 5 grams/liter begins to flow intothe vat at the rate of 2 liters/min. The thoroughly mixed content of the vat is drawn off at th erate of 3 liters/min. Which of the initial value problems below best describes the quantity ofsodium hydroxide, Q(t), that would be in the vat at time t, 0 < t < 80?(a) Q′= 10 −380 − tQ, Q(0) = 160.(b) Q′= 10 −3100Q, Q(0) = 160.(c) Q′= 10 −380 + tQ, Q(0) = 200.(d) Q′= 10 −3100 − tQ, Q(0) = 200.Page 5 of 11MATH 251 EXAMINATION I February 26, 20098. (5 points) Consider the autonomous equationy′= y3− 12y2+ 20y = y(y − 2)(y − 10).Suppose y(π2) = 10. What is limt→∞y(t)?(a) 0(b) 2(c) 10(d) ∞9. (5 points) Solve the following initial value problemy′′+ 3y′+ 2y = 0, y(1) = 1, y′(1) = 0.(a) 2e−(t−1)− e−2(t−1)(b) 2e−(t+1)− e−2(t+1)(c) e−t− e−2t(d) 3e2t− 2e3tPage 6 of 11MATH 251 EXAMINATION I February 26, 200910. (5 points) Given that y1= t2and y2= ln t are two solutions of a certain second order homo-geneous linear equation, y′′+ p(t)y′+ q(t)y = 0. All of the fu nctions below are also solutionsof the equation, EXCE P T(a) 0.(b) −3 ln t.(c) 2t2ln t.(d) t2− ln t.11. (5 points) Which of the equations below has y(t) = 5e−3tas a particular solution?(a) y′′+ 9y = 0(b) y′′− 6y′+ 9y = 0(c) y′′+ y′− 6y = 0(d) y′′− 2y′− 3y = 0Page 7 of 11MATH 251 EXAMINATION I February 26, 200912. (12 points) Consider the nonhomogeneous second order linear equation of the formy′′+ 2y′+ y = g(t).(a) (3 points) Find yc(t), the solution of its corresponding homogeneous equation.For each of the parts (b) through (d), ch oose from the list below the function that is themost suitable choice of the form of particular solution Y that you would use to solve thegiven equation using the Method of Undetermined Coefficients. DO NOT ATTEMPTTO SOLVE THE COEFFICIENTS.A. (At + B)e−tB. (At3+ Bt2)e−tC. Ae−tcos(t) + Be−tsin(t)D. Ae−tsin(t)E. (At2+ Bt + C)etcos(2t) + (Dt2+ Et + F )etsin(2t)F. At2etcos(2t) + Bt2etsin(2t)G. (At2+ Bt)e−tH. At3etcos(2t) + Bt3etsin(2t)I. At2etcos(2t)(b) (3 points) y′′+ 2y′+ y = e−tsin t(c) (3 points) y′′+ 2y′+ y = 2te−t(d) (3 points) y′′+ 2y′+ y = −6t2etcos(2t)Page 8 of 11MATH 251 EXAMINATION I February 26, 200913. (9 points) A mass-spring system is described by the equ ationu′′+ 6u′+ k u = 0.(a) (3 points) Find the value(s) of k that would make the system critically damped.(b) (3 points) If k = 25, what is the quasi-frequency of this mass-spring system?(c) (3 points) True or false: If k = 6, then there are some nonzero solutions of this mass-springsystem that will cross the equ ilibrium position more than ten times.Page 9 of 11MATH 251 EXAMINATION I February 26, 200914. (10 points) Solve the initial value problem:ty′+ 2y = cos t, y(π) = 0.Page 10 of 11MATH 251 EXAMINATION I February 26, 200915. (10 points) Given that y1(t) = t3is a known solution of the second order linear differentialequationt2y′′− 5ty′+ 9y = 0, t > 0.Find the general solution of th e equation.Page 11 of
View Full Document
Unlocking...