Question

Problem #2 Solve the initial value problem as follows: dzy 27+246X Calculate the value of d at the point x - 2, round-off the number to four figures and present it below (11 points): (your numerical result for the derivative must be written here) Also, you must provide some intermediate results obtained by you while solving the problem above: 1) The substitution used to solve the differential equation is as follows (mark a correct variant) (3 points): ye x= ye

Answer 1

27 x^2 d^2 y/dx^2 + 45 x dy/dx + 246 y = 0, y (1) = 0, y' (1) = 3, Now this is homogenous equation Put x = e^z, then the given equation reduces to (27 D (D - 1) + 45 D + 246) y = 0 D = d/dz y (0) = 0 y' (0) = z Rightarrow (27 D^2 - 27 D + 45 D + 246) y = 0 Rightarrow (27 D^2 + 18 D + 246) y = 0. Now Auxiliary equation is 27 m^2 + 18 m + 246 = 0 solving m_1 = -1/3 + 3: m_2 = -1/3 - 3 i. Therefore y (z) = e^- 1/3 z y (0) = 0 Rightarrow A = 0 Rightarrow y (z) = e^- 1/3 z . B sin 37 y' (z) = -1/3 e^- 1/3 z B sin 37 + e^- 1/3 z B cos 37 . z Therefore y' (0) = z Rightarrow z = 3 . B Rightarrow B = 1. Therefore y (z) = e^- 1/3 z . z sin 37 = 3 . e^- 1/3 z . sin 37. \r\n