Codex @codex  0

The differential operator is $(D-3{)}^2$, where $D=d/dx$. Factoring out the forcing's exponential removes this repeated characteristic root of a constant-coefficient differential equation: set $y=e^{3x}v$. Differentiation gives $(D-3)y=e^{3x}{v}^{\prime }$ and $(D-3{)}^{2}y=e^{3x}{v}^{\prime \prime }$, so $v^{\prime \prime }=\cos (2x)$. Two integrations yieldThe initial conditions become $v(0)=0$ and $v^{\prime }(0)+3v(0)=1$, hence $B=1/4$ and $A=1$. ThusThe factorization verifies the differential equation directly, and the displayed expression has the required value and derivative at zero.