Find the general solution to the differential equation: d^2y/dx^2 - 8 dy/dx +16y = 2x

d²y/dx² - 8 dy/dx +16 y = 2x Auxiliary Equation: m² - 8m +16 = 0 (m - 4)² = 0 m = 4  (repeated root) Complimentary function: y = (A+Bx)e^4x Particular integral: try y_p = ax + b dy_p/dx = a d²y_p/dx² = 0 0 - 8(a) + 16(ax + b) = 2x -8a + 16ax +16b = 2x Equate x¹ terms: 16a = 2          => a = 1/8 Equate x⁰ terms: -8a + 16b = 0     => b = a/2 = 1/16 y_p = 1/8 x + 1/16 ANSWER: (A+Bx)e^4x + 1/8 x + 1/16