解微分方程..

y" + 4y' + 3y = 5sin2x

Auxiliary equation (輔助方程)

k2 + 4k + 3 = 0

(k + 1)(k + 3) = 0

k = -1 or -3

Complementary solution:

yc = Ae-x + Be-3x

Let the particular solution, yp = Csin2x + Dcos2x

yp' = 2(Ccos2x - Dsin2x)

yp" = -4(Csin2x + Dcos2x)

So, -4(Csin2x + Dcos2x) + 4[2(Ccos2x - Dsin2x)] + 3(Csin2x + Dcos2x) = 5sin2x

(-C - 8D)sin2x + (8C - D)cos2x = 5sin2x

Comparing coefficients,

8C - D = 0, -C - 8D = 5

So, C = -1/13 , D = -8/13

Therefore, the general solution: y = yc + yp

= Ae-x + Be-3x - 1/13 sin2x - 8/13 cos2x

我把過程和答案都po在下面的網址喔!
http://img15.imageshack.us/img15/1030/5566h.gif