Chapter 8 - Linear Differential Equations of Order n - 8.10 Chapter Review - Additional Problems - Page 575: 17

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Given $F(x)=4x\sin x-3e^{-2x}$ Since, $(D^2+1)^2$ is the annihilator of $G(x)=3e^{-2x}$ Also, we have: $(D+2)$ is the annihilator of $H(x)=4x\sin x$.This implies that the $(D+2)$ is the annihilator of $4H(x)=4x\sin x$ So, $F(x)=4H(x)-G(x)$ Therefore, $(D+2)(D^2+1)^2$ is the annihilator of F(x).