McGill University math-263(2015-Fall): Dierential Equations for Engineers Written Assignment # 2 Due date: Tue. Oct. 19 Hand-in in class (Total 60 point) (1) (10 points) (a) Show that the identities of operators L1 = (aD + b)2 = a2 D2 + 2abD + b2 ; L2 = (aD + b)3 = a3 D3 + 3a2 bD2 + 3b2 aD + b3 L3 = (D + a) (D + b) = (D + b) (D + a) are valid, if a, b are constants. What are L1 , L2 , L3 , if a = 2, b = 3? (b) Do the above identities hold, when a = a(x), b = b(x) are not constants? What are L1 , L2 , L3 , if a = x, b = x2 ? (2) (10 points) Give a pair of functions {v1 (x) = x sin x; v2 (x) = x| sin x|} on the interval I = [, ]. (a) Determine whether these functions are linearly independent or linearly dependent. (b) Show that the Wronskian W (x) = W [v1 , v2 ] exists and calculate its value on the interval (I). (c) Prove that {v1 (x) = sin x; v2 (x) = sin2 x} cannot be a set of fundamental solutions of any second order linear dierential equation of 2-nd order On (I). (3) (10 points) Prove that the pair of functions {y1 (x) = sin x; y2 (x) = x sin x} on the interval I = (0, ) are linearly independent. Construct a linear dierential equation of order 2, for which {y1 (x) = sin x; y2 (x) = x sin x} is a set of fundamental solutions. (4) (10 points) Find the set of fundamental solutions by using the dierential operator method: [D2 (2 1)D + ( 1)]y = 0, t (0, ); and determine the values of , for which all solutions tend to zero as t , also determine the values of , for which all solutions become unbounded as t . 1 (5) (10 points) Solve the IVP with dierential operator method: [D2 + 5D + 6I]y = 0, y(0) = 2, y (0) = > 0. (a) Determine the coordinates (tm , ym ) of the maximum point of the solution as function of . (b) Determine the smallest value of for which ym 4. (c) Determine the behaviour of tm and ym as . (6) (10 points) Solve the following IVP by changing variable t = ex and using dierential operator method, t2 y + ty + y = 0, y(1) = 0, y (1) = . Determine the value of , with which the solution obtained satises the condition: as y(10) = 1. Markers information: Problem-(3): by Kleinman, Michael (260529700), michael.kleinman@mail.mcgill.ca Problem-(4): by Zhao, Shiyi (260524161), shiyi.zhao@mail.mcgill.ca Problem-(5): by Qin, Chuan (260562917), chuan.qin2@mail.mcgill.ca Problem-(6): by Kang, Clara (260558716), heng.kang@mail.mcgill.ca Problem-(1)-(2): by Xia, Tian (260512669) tian.xia3@mail.mcgill.ca McGill University math-263(2015-Fall): Dierential Equations for Engineers Written Assignment # 2 Due date: Tue. Oct. 19 Hand-in in class (Total 60 point) (1) (10 points) (a) Show that the identities of operators L1 = (aD + b)2 = a2 D2 + 2abD + b2 ; L2 = (aD + b)3 = a3 D3 + 3a2 bD2 + 3b2 aD + b3 L3 = (D + a) (D + b) = (D + b) (D + a) are valid, if a, b are constants. What are L1 , L2 , L3 , if a = 2, b = 3? (b) Do the above identities hold, when a = a(x), b = b(x) are not constants? What are L1 , L2 , L3 , if a = x, b = x2 ? (2) (10 points) Give a pair of functions {v1 (x) = x sin x; v2 (x) = x| sin x|} on the interval I = [, ]. (a) Determine whether these functions are linearly independent or linearly dependent. (b) Show that the Wronskian W (x) = W [v1 , v2 ] exists and calculate its value on the interval (I). (c) Prove that {v1 (x) = sin x; v2 (x) = sin2 x} cannot be a set of fundamental solutions of any second order linear dierential equation of 2-nd order On (I). (3) (10 points) Prove that the pair of functions {y1 (x) = sin x; y2 (x) = x sin x} on the interval I = (0, ) are linearly independent. Construct a linear dierential equation of order 2, for which {y1 (x) = sin x; y2 (x) = x sin x} is a set of fundamental solutions. (4) (10 points) Find the set of fundamental solutions by using the dierential operator method: [D2 (2 1)D + ( 1)]y = 0, t (0, ); and determine the values of , for which all solutions tend to zero as t , also determine the values of , for which all solutions become unbounded as t . 1 (5) (10 points) Solve the IVP with dierential operator method: [D2 + 5D + 6I]y = 0, y(0) = 2, y (0) = > 0. (a) Determine the coordinates (tm , ym ) of the maximum point of the solution as function of . (b) Determine the smallest value of for which ym 4. (c) Determine the behaviour of tm and ym as . (6) (10 points) Solve the following IVP by changing variable t = ex and using dierential operator method, t2 y + ty + y = 0, y(1) = 0, y (1) = . Determine the value of , with which the solution obtained satises the condition: as y(10) = 1. Markers information: Problem-(3): by Kleinman, Michael (260529700), michael.kleinman@mail.mcgill.ca Problem-(4): by Zhao, Shiyi (260524161), shiyi.zhao@mail.mcgill.ca Problem-(5): by Qin, Chuan (260562917), chuan.qin2@mail.mcgill.ca Problem-(6): by Kang, Clara (260558716), heng.kang@mail.mcgill.ca Problem-(1)-(2): by Xia, Tian (260512669) tian.xia3@mail.mcgill.ca \f