tapia jat4858 HW05 clark 52990 This print out should have 11 questions Multiple choice questions may continue on the next column or page find all choices before answering 1 and so 5 y1 x x 4x 4 x 2 At x 2 therefore 001 10 0 points If y1 is the particular solution of the differential equation y1 2 26 002 10 0 points Determine whether the differential equation xy ln x x5 y 0 is linear dy 2y 4x2 5 dx x which satisfies y 1 5 determine the value of y1 2 1 The equation is not linear 2 The equation is linear correct 1 y1 2 24 2 y1 2 23 3 The equation is neither linear nor nonlinear 3 y1 2 26 correct Explanation 003 10 0 points Solve the differential equation 4 y1 2 22 5 y1 2 25 6 t Explanation The integrating factor needed for the first order differential equation is du u 6 t dt t2 12t 2C correct 1 u 2 t 6 2 u t2 6t C t 6 After multiplying both sides of by 1 x we can thus rewrite the equation as 3 u t2 6t C 2 t 6 5 d y 4 dx x2 x2 4 u t2 6t 2C 2 t 6 Consequently the general solution of is given by 5 u t2 6t C t 6 e R 2 x dx e 2 ln x 1 2 x 2 y 5 4x C 2 x x where C is an arbitrary constant For the particular solution y1 the value of C is determined by the condition y 1 5 since y 1 5 5 9 C t 0 Explanation 004 10 0 points Use the Bernoulli s method to solve the y3 8 differential equation y y 4 x x 1 2 2 1 y Cx16 19x3 tapia jat4858 HW05 clark 52990 2 y Cx16 1 2 1 19x3 1 2 2 16 3 y Cx 19x3 1 2 2 16 4 y Cx correct 19x3 1 2 2 16 5 y Cx 19x3 2 resistance of R ohms The voltage drop Q across the capacitor is where Q is the C charge in coulombs so in this case KirchQ dQ hoff s Law gives R I E t But I C dt 1 dQ Q E t Suppose the so we have R dt C resistance is 10 the capacitance is 0 05 F a battery gives a constant voltage of 60 V and the initial charge is Q 0 0 C 0 05 F 10 Explanation Switch keywords 005 10 0 points The performance level of someone learning a skill P t is a function of the training time t dP and given by the differential equation dt k M P t where k is a positive constant Two new workers were hired for an assembly line Jim processed 27 units during the first hour and 43 units during the second hour Mark processed 32 units during the first hour and 48 units the second hour Estimate the maximum number of units per hour that each worker is capable of processing Assume that P 0 0 1 66 for Jim and 64 for Mark correct 60 V Find the charge and the current at time t 1 Q t 3 e 2t I t 6 1 e 2t 2 Q t 6 1 e 2t I t 3 e 2t 3 Q t 3 1 e 2t I t 6 e 2t correct 4 Q t 6 e 2t I t 3 1 e 2t 5 Q t 3 1 e 2t I t 6 e 2t Explanation 2 75 for Jim and 68 for Mark 3 53 for Jim and 80 for Mark 4 62 for Jim and 66 for Mark 5 68 for Jim and 70 for Mark 007 10 0 points Populations of aphids and ladybugs are modeled by the equations dA 2A 0 01AL dt dL 0 7L 0 0001AL dt Explanation 006 10 0 points The figure shows a circuit containing an electromotive force a capacitor with a capacitance of C farads F and a resistor with a Find an equilibrium solution 1 A 7000 L 200 correct 2 A 6000 L 200 tapia jat4858 HW05 clark 52990 3 A 4000 L 100 3 A 5000 L 150 correct 4 A 7000 L 0 4 A 5000 L 350 5 A 9000 L 200 5 A 8000 L 200 3 Explanation Explanation 008 10 0 points Populations of aphids and ladybugs are modeled by the equations 010 10 0 points Populations of aphids and ladybugs can be modeled with a Lotka Volterra system as follows dA 5A 0 01AL dt dL 0 4L 0 0001AL dt dL dA 0 04A 0 00001AL Find an expression for dL 1 dA 2 3 4 5 5A 0 01AL 2 0 04L 0 00001AL dL dA 5A 0 01AL dL 0 4L 0 0001AL correct dA 5A 0 01AL 5L 0 01AL dL dA 0 4L 0 0001AL dL 5A 0 01AL dA 0 4L 0 0001AL Explanation dA 5A 1 0 0001A 0 01AL dt dL 0 7L 0 0001AL dt dL dA 5A 1 0 0001A 0 01AL dL 1 dA 0 07L 0 00001AL Find an expression for 2 dL 0 7L 0 0001AL dA 5A 1 0 0001A 0 01AL 3 dL 0 07L 0 00001AL dA 5A 1 0 0001A 0 01AL 4 5A 1 0 0001A 0 01AL dL dA 0 7L 0 0001AL 5 dL 0 7L 0 0001AL cordA 5A 1 0 0001A 0 01AL rect 009 10 0 points Populations of aphids and ladybugs can be modeled with a Lotka Volterra system as follows dA 3A 1 0 0001A 0 01AL dt dL 0 5L 0 0001AL dt Explanation 011 10 0 points Populations of aphids and ladybugs can be modeled with a Lotka Volterra system as follows 1 A 7000 L 50 dA 2A 1 0 0001A 0 01AL dt dL 0 5L 0 0001AL dt 2 A 7000 L 150 Suppose that at time t 0 there are 1000 aphids A and 200 ladybugs L Find the equilibrium solutions tapia jat4858 HW05 clark 52990 Choose the corresponding phase trajectory 1 L 400 200 A 2000 4000 6000 2 L 150 100 50 A 2000 correct Explanation 4000 4