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[fortpost]

15. y''+y=cosx+cos2x, y(0)=1, y'(0)=0

k²+1=0 -> k₁=i, k₂=-i

y₀=C₁cosx+C₂sinx
y=C₁(x)cosx+C₂(x)sinx

C'₁(x)cosx+C'₂(x)sinx=0
-C'₁(x)sinx+C'₂(x)cosx=cosx+cos2x
C'₁(x)=-sinxcosx-sinxcos2x
C'₂(x)=cos²x+cosxcos2x
C₁=∫(-sinxcosx-sinxcos2x)dx=1/2cos²x+1/6cos3x-1/2cosx+C₁₁
C₂=∫(cos²x+cosxcos2x)dx=x/2+1/4sin2x+1/2sinx+1/6sin3x+C₁₂

y=(1/2cos²x+1/6cos3x-1/2cosx+C₁₁)cosx+(x/2+1/4sin2x+1/2sinx+1/6sin3x+C₁₂)sinx
y'=-1/2sinxcos²x-1/3sin3xcosx+1/3sinxcos3x+sinxcosx-C₁₁sinx+1/2sinx+1/2xcosx-1/2sin³x+C₁₂cosx

y(0)=1/2+1/6-1/2+C₁₁=1 -> C₁₁=5/6
y'(0)=C₁₂=0 -> C₁₂=0

y=(1/2cos²x+1/6cos3x-1/2cosx+5/6)cosx+(x/2+1/4sin2x+1/2sinx+1/6sin3x)sinx