tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 10

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The general solution of the differential equation is thus y C1 cos x c2 sin x x. Applying the two initial conditions gives us the equations Cl 1 C2 1 0. The solution subject to the initial conditions is y cos x sin x x. Solution Solve x2y z x xy x y x x. The homogeneous equation is x2y x xy x y x 0. Substituting y xx into the homogeneous differential equation yields x2A A 1 xĂ-2 xAxĂ xx 0 A2 2A 1 0 A 1 2 0 A 1. The homogeneous solutions are yi x y2 x log x. The Wronskian of the homogeneous solutions is Tz r _1__1 x x log x W log 1 log x x x log x x log x x. 1134 Writing the inhomogeneous equation in the standard form y x - 1 y x x 1 X 1 y x -. xx Using variation of parameters to find the particular solution x i g dx x log x i dx xx x 2 log2 x x log x log x - x log2 x. Thus the general solution of the inhomogeneous differential equation is y c1x c2x log x -x log2 x. Solution 1. First we find the homogeneous solutions. We substitute y eAx into the homogeneous differential equation. y y 0 A2 1 0 A 1 y e We can also write the solutions in terms of real-valued functions. y cos x sin x .