Question

4. a) Find the general solution of the Cauchy-Euler equation 4x3y" - 4x2y"+3xy 0 b) Use the variation of parameters to find the general solution of 4x3y"-4x2y, + 3x/ = 6x7/2

Answer 1

4 x^3y''' - 4 x^2y'' + 3 xy' = 0 - (1) is a Cauchy - euler D.E let x = e^z z = ln x xD = D, x^2 D^2 = D_1 (D_1 - 1), x^3 D^3 = D_1 (D_1 - 1) (D_1 - 2) (1) Implies [4 D_1 (D_1 - 1) (D_1 - 2) - 4 D_1 (D_1 - 1) + 3 D_1] y = 0 Implies [4 D_1^3 - 8 D_1^2 - 4 D_1^2 + 8 D_1 - 4 D_1^2 + 4 D_1 + 3 D_1] y = 0 Implies (4 D_1^3 - 16 D_1^2 + 15 D_1) y = 0 - (2) A.E of (x) s f (m) = 0 Implies 4 m^3 - 16 m^2 + 15 m = 0 Implies (4 m^2 - 16 m + 15) m = 0 Implies m = 0, 4 m^2 - 10m - 6m + 15 = 0 2m (2m - 5) - 3 (2m - 5) = 0 (2m - 5) (2m - 3) = 0 m = 3/2, 5/2 Therefore general solution of (2) is y = c_1 + c_2 e^3z/2 + c_3 e^5z/2 y = c_1 + c_2 x^3/2 + c_3 x^5/2. (b) 4 x^3y''' - 4 x^2y''