MA26600 Final Exam: Key Concepts in Differential Equations
School
Epic Charter School**We aren't endorsed by this school
Course
EPIC ANSWERS
Subject
Mathematics
Date
Dec 10, 2024
Pages
13
Uploaded by UltraAtom7917
MA26600 Final ExamGREEN VERSION 01NAME:INSTRUCTOR:SECTION/TIME:•You must use a#2 pencilon the mark-sense answer sheet.•Fill in theten digit PUID(starting with two zeroes) and yourNameand blacken inthe appropriate spaces.•Fill in the correctTest/Quiz number(GREEN is01, ORANGE is02)•Fill in thefour digit section numberof your class and blacken the numbers belowthem. Here they are:0010MWF10:30AMGayane Poghotanyan0011MWF9:30AMGayane Poghotanyan0022MWF11:30AMGayane Poghotanyan0034MWF3:30PMGuang Yang0035MWF2:30PMGuang Yang0046MWF12:30PMHeejong Lee0061MWF12:30PMJiahao Zhang0062MWF9:30AMArun Debray0071MWF10:30AMArun Debray0072TR12:00PMXingshan Cui0083TR10:30AMXingshan Cui0084MWF1:30PMPing Xu0095MWF1:30PMJiahao Zhang0096MWF1:30PMKrishnendu Khan0107MWF12:30PMKrishnendu Khan0108MWF2:30PMShuyi Weng0109MWF3:30PMShuyi Weng0110MWF9:30AMJakayla Robbins0111MWF8:30AMJakayla Robbins0112MWF11:30AMPing Xu0113MWF12:30PMPing Xu0114TR1:30PMChristian Noack•Sign the mark-sense sheet.•Fill in your name and your instructor’s name and the time of your class meeting on theexam booklet above.•There are 20 multiple-choice questions, each worth 10 points.Blacken inyour choiceof the correct answer in the spaces provided for questions 1–20 in the answer sheet. Doall your work on the question sheets, in addition, alsoCircleyour answer choice for eachproblem on the question sheets in case your mark-sense sheet is lost.•Show your work on the question sheets.Although no partial credit will be given, anydisputes about grades or grading will be settled by examining your written work on thequestion sheets.•No calculators, books, electronic devices, or papers are allowed.Use the back ofthe test pages for scratch paper.•Pull off thetable of Laplace transformson the last page of the exam for reference. Donot turn it in with your exam booklet at the end.1
1.Find a general solution of the following differential equation:dydx=x+ 4√xy,(x >0, y >0)A.12x3/2+ 8x1/2−23y2/3=CB.32x2/3+ 8x1/2+32y2/3=CC.y=x+ 12x−1+CD.y= (x3/2+ 12√x)2/3+CE.23x3/2+ 8x1/2−23y3/2=C2.Find a general solution to the following differential equationdydx+ 3y= 3x2e−3x+ 2xe−3x.A.y= (x3+x2)e−3x+CB.y= (x3+x2+C)e−3xC.y= (3x2+ 2x+C)e−3xD.y=−x2+43x+49e−3x+CE.y= (x3+x2)e−3x+Ce3x2
3.Which of the following is an implicit solution to the differential equation(2x+ 7y)dx+ (7x+ 8y)dy= 0?A.x2+ 7xy+ 4y2=CB.x2−7xy−4y2=CC.x2+ 4y2=CD.x2−4y2=CE.x3−7xy+ 4y2=C4.Which of the following figures sketches typical solution curves of the differential equationdydx= 2y2(y−1)?01234−2−1012A.01234−2−1012B.01234−2−1012C.01234−2−1012D.01234−2−1012E.3
5.Solve the initial value problemy′′−4y′+ 4y= 0,y(0) = 12, y′(0) =−3.A.y(t) = 10e2t−25te2tB.y(t) = 12e2t−27te2tC.y(t) =−15e2tD.y(t) = 5e2t+ 7e−2tE.y(t) = 27e2t−15e−2t6.The nonhomogeneous differential equationx2y′′−4xy′+ 6y=x3,x >0has complementary solutionyc(x) =c1x2+c2x3.Using the method of variation ofparameters, the particular solutionyp(x) is given byA.yp(x) = 4x5−6x4B.yp(x) =x5/3−x4C.yp(x) =x3−x2lnxD.yp(x) =x5/6E.yp(x) =x3(lnx−1)4
7.Solve the initial value problemy′′−3y′−4y= 3e2x,y(0) =52, y′(0) = 1.A.y(t) =32e−x+ 2e4x−e2xB.y(t) = 4ex−e−4x−12e2xC.y(t) =e−x+e4x+12e2xD.y(t) = 2e−x+e4x−12e2xE.y(t) =72ex−2e−4x+e2x8.A spring-mass system set in motion is determined by the initial value problemu′′+ 100u= 0,u(0) = 2,u′(0) =−10.What is the amplitude of the motion?A.√5B.√2C. 10D.√104E.1105
9.A fish tank contains 10 gallons of a salt solution with a concentration of 3 grams of saltper gallon. A salt solution with a concentration of 6 grams/gallon is added to the tankat a rate of 2 gallons per minute. At the same time, the solution is drained from the wellmixed tank at a rate of 2 gallons per minute. How many grams of salt are in the tankafter 10 minutes?A. 30 + 12e−5B. 30 + 30e−10C. 120−90e−2D. 60−30e−2E. 6010.Determine the appropriate form for a particular solution ofy(3)+ 4y′′+ 4y′= 4e−2xx.A.Ae−2xx3B.Ae−2xx3+Be−2xx2C.Ae−2xx3+Be−2xx2+Ce−2xx+De−2xD.Ae−2xx+Be−2x+CE.Ae−2xx2+Be−2xx6
11.Which of the following functions is a solution ofy(4)−16y= 0?A.y= 3e2t−e−2t+ 5 cost−2 sintB.y= 16et−cos 4t−sin 4tC.y= 5e2t−e−2t+ 7 cos 2t−3 sin 2tD.y= (2t+ 1)e2t+ (3t−4)e−2tE.y= (t3+ 3t2−5t+ 1)e2t12.Consider the linear systemx′=416−1x.What is the phase portrait at the origin?A. Saddle pointB. Spiral sinkC. Spiral sourceD. Nodal sourceE. Center7
14.IfAis a 2 x 2 real-valued matrix with eigenvaluesλ1=−3 andλ2= 4 with correspondingeigenvectorsv1=1−1andv2=3−2, find the matrix exponentialeAt.A.4e4t−3e−3t2e4t−2e−3t3e−3t−3e4t4e−3t−4e4tB.e−3t3e4t−e−3t−2e4tC.e−3t00e4tD.3e4t−2e−3t3e4t−3e−3t2e−3t−2e4t3e−3t−2e4tE.−2e−3t−3e−3te−3te4t9
15.Determine the appropriate form for a particular solutionxp(t) to the nonhomogeneouslinear system:x′(t) =1221x(t) +−e−t2etA.xp(t) =ate−t+betB.xp(t) = (at+b)e−t+ (ct+d)etC.xp(t) =ae−t+betD.xp(t) =ae−t+ (bt+c)etE.xp(t) = (at+b)e−t+cet16.After applying the Laplace transform to the differential equationx′′+ax=b t3,x(0) = 1, x′(0) = 0,one obtains thatX(s) =L{x(t)}is given by the algebraic formulaX(s) =1s+1s6.Find the values ofaandb.A.a= 0,b=13B.a= 0,b=16C.a=−1,b=12D.a= 1,b=16E.a= 1,b= 310
19.Find the Laplace transform off(t) =(t,if 3≤t <5,0,otherwise.A.e−3s1s2+3s−e−5s1s2+5sB.e−3s1s2+3s+e−5s1s2+5sC.e−3ss2−e−5ss2D.e−3ss2+e−5ss2E.e−2ss220.LetF(x) =L{f(t)}be the Laplace transform of the functionf(t) =Zt0e−τcos(2τ) sin(t−τ)dτ.What is the value ofF(1)?A.F(1) =316B.F(1) =110C.F(1) =18D.F(1) = 4E.F(1) =13212