y'' - 2y' + y = 3sin(2x)

y'' - 2y' + y = 3sin(2x)

Step1: 특성방정식

r² - 2r + 1 = 0

(r - 1)² = 0

r = 1 (중근)

일반해 y_h = (c₁ + c₂x)e^x

Step2: 미정계수법

y_p = A cos(2x) + B sin(2x)

y_p' = -2A sin(2x) + 2B cos(2x)

y_p'' = -4A cos(2x) - 4B sin(2x)

원 식에 대입

y'' - 2y' +y = 3sin(2x)

(-4A cos(2x) - 4B sin(2x) ) - 2(-2A sin(2x) + 2B cos(2x)) +(A cos(2x) + B sin(2x)) = 3sin(2x)

(-3A - 4B) cos(2x) + (4A - 3B - 3) sin(2x) = 0

(-3A - 4B) = 0

(4A - 3B - 3) = 0

A = -4B/3

A(-4B/3) - 3B = 3

B = -9/25, A = (-4/3)(-9/25) = 12/25

y = (c₁+c₂x)e^x + (12/25) cos(2x) - (9/25) sin(2x)