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[fortpost]

16. y'''-3y''-y'+3y=2cosx+2

k³-3k²-k+3=0
k²(k-3)-(k-3)=0
(k²-1)(k-1)=0
k₁=-1, k₂=1, k₃=3

y₀=C₁e^-x+C₂e^x+C₃e^3x
y=C₁(x)e^-x+C₂(x)e^x+C₃(x)e^3x

C'₁(x)e^-x+C'₂(x)e^x+C'₃(x)e^3x=0
-C'₁(x)e^-x+C'₂(x)e^x+3C'₃(x)e^3x=0
C'₁(x)e^-x+C'₂(x)e^x+9C'₃(x)e^3x=2cosx+2

C'₁=1/4(cosx+1)e^x
C'₂=-1/2(cosx+1)e^-x
C'₃=1/4(cosx+1)e^-3x

C₁=∫1/4(cosx+1)e^xdx=1/4(1/2e^xcosx+1/2e^xsinx+e^x+C₁₁)
C₂=∫-1/2(cosx+1)e^-xdx=-1/2(-1/2e^-xcosx+1/2e^-xsinx-e^-x+C₂₁)
C₃=∫1/4(cosx+1)e^-3xdx=1/4(-3/10e^-3xcosx+1/10e^-3xsinx-1/3e^-3x+C₃₁)

y=3/10cosx-1/10sinx+2/3+1/4C₁₁e^-x-1/2C₂₁e^x+1/4C₃₁e^3x