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)e4x Particular integral: try yp = ax + b dyp/dx = a d2yp/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 yp = 1/8 x + 1/16 ANSWER: (A+Bx)e4x + 1/8 x + 1/16
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