Math 217 Final Exam Page 1 Name ID Section This exam has 16 multiple choice questions Important No graphing calculators For the multiple choice questions mark your answer on the answer card Show all your work for the written problems You will be graded on the ease of reading your solution You are allowed a 3 5 note card for the exam Function Transform Function f 0 t sF s f 0 1 f 00 t s2 F s sf 0 f 0 0 t F s s tn eat f t F s a ta u t a f t a Z t f g t f g t d e as F s 1 t F s G s cos kt F 0 s sin kt Rt 0 f d 0 tf t 1 f t t f t period p u t a eat tn eat R F d cosh kt e st f t dt sinh kt s 1 1 e ps Z p 0 e as 1 sin kt kt cos kt s 2k 3 1 t sin kt s a 2k n 1 sin kt kt cos kt n 1 s a 2k sin A B sin A cos B cos A sin B cos A B cos A cos B sin A sin B 2 cos A cos B cos A B cos A B 2 sin A sin B cos A B cos A B 2 sin A cos B sin A B sin A B Transform 1 s 1 s2 n sn 1 a 1 sa 1 1 s s s2 k 2 k 2 s k2 s 2 s k2 k 2 s k2 1 2 s k 2 2 s 2 s k 2 2 s2 s2 k 2 2 Math 217 Final Exam Page 2 1 Solve the differential equation dy 3x2 y y dx y 1 2 Find y 2 and select the closest answer a 0 b 10 c 25 d 50 e 100 f 200 g 500 h 800 CORRECT i 1000 j 1500 k 2500 Solution This equation is separable and has solution y Cex 3 x 3 y 2ex x y 2 806 86 Math 217 Final Exam Page 3 2 Solve the initial value problem 1 y 0 y 2 sin x2 x y 0 What is y 2 5 a 1 0 b 0 9 c 0 8 CORRECT 0 79977979 d 0 7 e 0 6 f 0 5 g 0 4 h 0 3 i 0 2 j 0 1 k 0 0 Solution First order linear equation C cos x2 x 1 cos x2 y x y Math 217 Final Exam Page 4 3 Solve the initial value problem xy 2 y 0 x3 y 3 y 1 0 What is y 2 a 5 b 2 5 c 0 d 2 5 CORRECT 2 553 e 5 f 7 5 g 10 h 12 5 i 15 j 17 5 k 20 Solution Homogeneous let v xy 1 v 2 v 3 v 3 ln x 3 y x 3 ln x xv 0 v Math 217 Final Exam Page 5 4 Solve the initial value problem yexy xexy 1 y 0 0 y 0 1 What is x when y 0 5 a 0 6 b 0 7 c 0 8 CORRECT 0 8109 d 0 9 e 1 0 f 1 1 g 1 2 h 1 3 i 1 4 j 1 5 k 1 6 Solution Exact exy y C exy y 2 1 ex 2 2 x 2 ln 3 2 2 Math 217 Final Exam Page 6 5 Find the equilibrium solutions of the autonomous equation y 0 y 3 y 2 2y and determine which equilibria are stable a Stable y 2 y 1 Unstable y 0 b Stable y 2 y 2 Unstable y 0 c Stable y 1 y 1 Unstable y 0 d Stable y 1 y 2 Unstable y 0 e Stable y 0 y 3 Unstable y 1 f Stable y 1 y 3 Unstable y 2 g Stable y 0 Unstable y 2 y 1 CORRECT h Stable y 0 Unstable y 2 y 2 i Stable y 0 Unstable y 1 y 1 j Stable y 1 Unstable y 0 y 3 k Stable y 2 Unstable y 1 y 3 Solution y y y 1 y 2 Equilibrium solutions are y 2 y 0 y 1 Only y 0 is stable Math 217 Final Exam Page 7 6 Use Runge Kutta with step size h 0 5 to estimate y 1 y 0 x2 xy 2 y 0 1 a 1 9 b 2 0 c 2 1 d 2 2 e 2 3 f 2 4 g 2 5 h 2 6 i 2 7 CORRECT j 2 8 k 2 9 Solution n xn yn k1 k2 k3 k4 k 0 0 0 1 0 0 0 3125 0 3531 0 9421 0 3789 1 0 5 1 1894 0 9574 2 0936 2 7629 7 6094 3 0466 2 1 0 2 7127 Math 217 Final Exam Page 8 7 Solve the initial value problem y 00 2y 0 2y 0 y 0 0 y 0 0 1 What is y 1 a 1 1 b 1 6 c 1 9 d 2 3 CORRECT e sin 1 2 28735529 e 2 5 f 2 7 g 2 9 h 3 1 i 3 5 Solution y ex c1 cos x c2 sin x y ex sin x Math 217 Final Exam Page 9 8 Which of the following are solutions to the differential equation y 00 2y 0 3y 2 cos x 6 sin x I y sin x cos x II y ex sin x cos x III y e 3x ex IV y e 3x ex sin x a I only b II only c IV only d I and II only CORRECT e I and IV only f II and III only g I II and III only h I II and IV only i II III and IV only j I II III and IV k Some other answer Solution General solution is y cos x sin x c1 ex c2 e 3x Math 217 Final Exam Page 10 9 For what value of will the following mass spring system have resonance 2 x00 x sin 2t a 0 b 1 8 c 1 4 d 1 2 CORRECT e 1 2 f 2 g 2 h 4 i 8 j No value of will cause resonance Solution The solution to the homogeneous problem is x c1 cos t c2 sin t Therefore resonance will occur then 1 2 or when 12 Math 217 Final Exam Page 11 10 Consider the differential equation modelling a mass spring system 31x00 16x0 2x 0 a This equation is an example of resonance b This equation is an example of overdamping CORRECT c This equation is an example of underdamping d This equation is an example of critical damping e None of the above Solution The equation …
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