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Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 6

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We make the substitution u eA- A2 1 0 A i A set of linearly independent solutions for u is e e . Since e A cos Ẹ 2 an sin Ẹ 12 another linearly independent set of solutions is cos sin . The general solution for y x is y x Cl cos ln x c2 sln ln x . Solution 17.12 Consider the differential equation x2y 2xy 2y 0. With the substitution y xx this equation becomes A A 1 2A 2 0 A2 3A 2 0 A 1 2. The general solution is then y C1x c2x2. 974 Solution 17.13 We note that xy v 1V 0 x is an Euler equation. The substitution y xx yields A3 - 3A2 2A A2 - A A 0 A3 - 2A2 2A 0. The three roots of this algebraic equation are A 0 A 1 i A 1 1 The corresponding solutions to the differential equation are y x0 y x1 1 y x1-1 y 1 y x e1 ln x y x e-1 ln x We can write the general solution as y c1 c2x cos ln x c3 sin ln x . Solution 17.14 We substitute y xx into the differential equation. x2y 2a 1 xy by 0 A A - 1 2a 1 A b 0 A2 2aA b 0 A -a v a2 - b .