MATH 308 Differential Equations Test Instructions and Guidelines

Test TA Spring Semester 2024 AI‘M TEXAS A&M MATH 308 Differential Equations UNIVERSITY Dr. Minchul Kang Texas A&M University | Department of Mathematics | 335B Blocker Building | College Station | TX 77840 | grcorg@tamu.edu Direction o Do not turn to the next page until instructed. o Please turn off all mobile phones, calculators, and other electronic device, and stow those away in your bags. o Print your name and UIN below. o0 Your desk may only contain pens, pencils, erasers, and student IDs during the test. No calculators, no notes, no formula sheets, no extra scratch paper are allowed. o In part I (Multiple choice questions), print your answer clearly and legibly in the box provided. o In part IT (Short answer questions), ensure that your final response is typed clearly and legibly in the provided answer box. o In part III (Work out questions), show your work to earn credits for work out problems: merely having final answers does not suffice. o There are 10 pages in this test set, including a scratch paper on the last page. Carefully separate the scratch paper from the test booklet. Do not turn in this scratch paper. Any work shown on the scratch paper will neither be accepted nor graded. Name UIN Section ] 507 ) ] 515 Last name Fist name THE AGGIE HONOR CODE "An Aggie does not lie, cheat, or steal or tolerate those who do.” Signature: Test IA | MATH 308 Differential Equations [1/10]

] Part I. Multiple choice (3 ptsx6) Problem 3. Which of the following initial value prob- lems may not have the unique solution on [0, c0) X [0, 00)? Print your answer in the box provided. A .y =uzsiny, y(0) =0 B .y =ysinz, y(0) =0 Problem 1. Choose a correct classification of the fol- C . ! — O prny O * lowing differential equation y' =y, y(0) I J— T e + T2Y? Usy + YUz, = sin(x + y) D .y =yyz, y(0) =0 E . None of above A . Nonhomogeneous 4th order nonlinear PDE B . Nonhomogeneous 2nd order nonlinear PDE [Answer] C . Homogeneous 4th order nonlinear PDE D . Homogeneous 2nd order nonlinear PDE E . None of above * [Answer| Problem 4. Find a differential equation of the direc- tion fields. Problem 2. Which statement is true for solutions for the direction fields? g Ry R o e el A TR A A —h T Y B S UNONCONNNON N NN N T T T Y N T T A TR TR TR TR T T T T T NN NN NN NN OO N N N N N Sa AW R M S A W e s TR TR TR TR A TR T T T T e e e e e g B i § B e e i P e e e e e e s i e e Sl S S B e e e i e AT e e A B e . b I e P e e e T ' B i I R I S e i R I S S S D il e d d e g D i g g de e e A g 1 AAAA AT AT AT FaxmaZanmaN OV e N R R R R R R FAAAAIAATARAAAA AT TS e N R R e Ll ' ' VDI IYD VNP VNS ;;;:j::::; :;?:}i:??t FAPIIIPII NP AP PP AIAS Y ISNNN DR R R RS PPIPPIIPIIRIPP PRI AN I SNV ENRRE SRR AR PPPIPPAPPPIRL PP I I SN \zz7xx:'xt PPAPPIIPPMELPLLPLLLLLN I PN SN SRR PIPPIPAIIIRIPIIPIPY R N R S | EEERE RSN EEN R RN NN R R R RNy, RN AN NN A Ay =(@+D)y+1) — J— A . For x < 0 all solutions are decreasing. B.y=(@-1)y+1) . . r_ Y B . lim,; , f(x) =1 for all solutions f(z). * C.y=@+Dly-1) ,_ —_— —_ C . For y > 1 all solutions are increasing. D.y=(@-1-1) D .y =0forz=0. E . None of above E . None of above. [Answer] [Answer| Test IA | MATH 308 Differential Equations [2/10]

Problem 5. What is a possible substitution to solve xy = (1+ 22 — /Y)Y ] Problem 6. For y' = 32 — 4 , which of followings is NOT true? A u=12 — 1 % B.u-\/y C.u=y D.u=1+2>—-2y E.u=231 y [Answer| A . y =2 is a steady state solution. B . y = —2is an equilibrium. C . The equation has a unique solution for y(0) = 0. D . The equation has two solution for y(0) = 2.* E . None of above [Answer]| Test 1A | MATH 308 Differential Equations 3/10]

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Part II. Work out questions (10 ptsx5) Show your complete work to earn credits: merely having final answers does not suffice. In accurate Problem 7. ] Identify an equation category, outline a solution method, and solve the equation. v =20 /T statements will be marked incorrect even if they || Category: Separable achieve correct results. (1 pts) Method: Separation of variables For Problem 7 to 11, choose the equation category (2 pts) from Solution: y = sin (332 + C’) o 2nd order linear w/ constant coefficients (7 pts) o Bernoulli’s equation o Cauchy-Euler equation o Exact (w/ or w/o integration factor) o Homogeneous with degree zero o Integrable o Linear 1st order o Linear composite o Separable Choose the solution method from o Characteristic equation o Direct integration o Integrating factor o Partial integrations o Separation of variables o Substitution v = ax + by + ¢ o Substitution u = y* o Substitution y = ux [Show your work here.] y =2xy/1—y? 1 dy = 2xdzx V1 —y? arcsiny = 22 + C y = sin(z? + C) Test 1A | MATH 308 Differential Equations [4/10]

] Problem 8. Identify an equation category, outline a | Problem 9. Identify an equation category, outline a solution method, and solve the equation. solution method, and solve the equation. vy —y=a?sinz, y(%)=0. (6z + 4y)dx = (8y® — 4x)dy. Category: 1st order linear Category: Exact (1 pts) (1 pts) Method: Integrating factor Method: Partial integration (2 pts) (2 pts) Solution: Y = —T COST Solution: 3x2 4 4y — 2y* = C (7 pts) (7 pts) [Show your work here.] [Show your work here.] zy —y=2?sinz (62 + 4y)dx + (4 — 8y*)dy = 0 1 — 3 — Sy =xsinz (6x + 4y), = 4vs. (4o — 8y~), =4 x fo =6+ 4y f(_l)dx L 2 IF =l 9% = = f =32 +4xy + C(y) ’ _ ! _ _ 3 (£) =sna fy =+ C'y) = 42— 8y o Cly)=-24*+C ;:—cosx—l—C' F=322 4 day— 2t 4 C 0 y =—cos(n/2)+C;C =0 Yy = —XCoSx Test IA | MATH 308 Differential Equations [5/10]

] Problem 10. Identify an equation category, outline a | Problem 11. Identify an equation category, outline a solution method, and solve the equation. solution method, and solve the equation. y//_2y/_|_y:O y//+2y/+2y20 Category: 2nd order linear w/ constant Category: 2nd order linear w/ constant coefficients coefficients (1 pts) (1 pts) Method: Characteristic equation Method: Characteristic equation (2 pts) (2 pts) Solution: y= e+ tet Solution: y=-e" " (Cycost+ Cysint) (7 pts) (7 pts) [Show your work here.] [Show your work here.] Char. Eq. A2 —2\+1=0= )\ = 1 (repeated) Char. Eq. M2 +22+2=0=)\= -1+ y = Cre’ + Cytel y=¢e¢ " (Cicost + Cosint) Test IA | MATH 308 Differential Equations [6/10]

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] Part 1II. Multistep Workout questions Step 2 (6 pts): Use reduction of order to find a (16ptsx2) second solution ys(x). Show your complete work to earn credits: merely having final answers does not suffice. In accurate statements will be marked incorrect even if they | From the order reduction formula for y” + p(x)y + [Show your work here.] achieve correct results. g(x)y =0,0r y’ — 1y + Ly = 0. e—fp(x) Problem 12. Find the general solution to Cauchy Eu- Y2 = Y1 / le“ ler’s equation following suggested steps 1 =z a:2€_f1/tdtda: 22y —xy +y=0forxz>0. - =zxlnx Step 1 (5 pts): By making a guess y(x) = «” for some 7, find a solution y;(x) to the equation (i.e. find r by plugging x" into the equation). [Show your work here.] O=r(r—1)—r+1 =72 —2r+1 = (r—1)*? oY1 =X Test IA | MATH 308 Differential Equations [7/10]

Step 3 (5 pts): Show that y; and y» form the complete set of solution (i.e. independent). [Show your work here.] z xlnzx Wiyt y2) = 1 lnzx+1 Inz+xz—xlnz =z >0 ] Problem 13. Following suggested steps to determine the general solution to vy’ + 2y’ — 3y = 4e? Step 1 (5 pts): Solve the associated homogeneous equation to find y.. [Show your work here.] y'+ 2y —3y =0 A +2X0-3=0 A+3)(A=1)=0 Yo = Cret + Coe™ 3t Test 1A | MATH 308 Differential Equations 8/10]

] Step 2 (7 pts): Find y, by the method of unde- termined coefficients. Step 3 (4 pts): Find the general solution to [Show your work here.] yp(t) = Ate® (- €' is part of y.) yp(t) = A(1 +t)e* Yy (t) = A(2 +t)e* Plugging into the differential equation, A2+ t)et + 2A(1 + t)et — 3Ate! = 4et A(142 = 3)te’ + A(2 + 2)e’ = 4e? 4Aet = 4et A=1 L yp(t) - tet v’ + 2y — 3y = 4t [Show your work here.] Yo = Crel 4+ Coe™3t 4 tet | | Mo SA LA Total Score Test 1A | MATH 308 Differential Equations [9/10]

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] Scratch paper Carefully separate the scratch paper from the test booklet. Do not turn in this scratch paper. Any work shown on the scratch paper will neither be accepted nor graded. Test IA | MATH 308 Differential Equations [10/10]

Test1_24S_a_sol

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