Question

Homework: Section 6.2 Score: 0 of 1 pt 6.2.19 6 of 6 (1 complete) Solve the given initial value problem. y",-2y',-25y' + 50y-o y(0)-2, y(0)- -55, y' (0)--55 yk)

Answer 1

"Given y""' - 2y"" - 25 y' + 50 y = 0 y(0) = 2, y'(0) = -55 y'(0) = -55 The A.E f(x) = 0 Implies m^3 - 2m^2 - 25 m + 50 = 0 put m = 2 Implies 8 - 8 - 50 + 50 = 0 Implies mm^2 - 25 = 0 Implies m^2 = 25 Implies m = plusminus 5 m = 2,-5,5 Implies -5,2,5 y_c = C_1 e^-5x + C_2 e^2x + C_3 e^5x The General solution y = C_1 e^-5x + C_2 e^2x + C_3 e^5x ....(1) y(0) = C_1 e^-5(0) + C_2 e^2(0) + C_3 e^5(0) 2 = C_1 + C_2 + C_3 Implies C_1 + C_2 + C_3 = 2 .....(2) D .w r to x in eq (1) y'(x) = C_1 e^-5x (-5) + C_2 e^2x (2) + C_3 e^5x (5) ...(3) y'(0) = C_1 e^-5(0) (-5) + 2 C_2 e^2(0) + 5 C_3 e^5(0) Implies -55 = -5C_1 + 2C_2 + 5C_3 Implies -5C_1 + 2C_2 + 5C_3 = -55 D w r to x in eq (3) y"" (x) = -5 C_1 e^-5x (-5) + 2C_2 e^2x (2) + 5C_3 e^5x (5) y"" (x) = 25 C_1 e^-5x + 4 C_2"