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Question Number 160752 by Eric002 last updated on 05/Dec/21

$$solve$$$$\frac{d^{2}y}{{dx}^2}-y=x^{2}sin3x$$

Answered by qaz last updated on 06/Dec/21

$${\mathrm{y}}^{″}-\mathrm{y}={\mathrm{x}}^2\sin 3\mathrm{x}$$$${\mathrm{y}}_{\mathrm{p}}=\frac{1}{{\mathrm{D}}^2-1}{\mathrm{x}}^2\sin 3\mathrm{x}$$$$=\frac{1}{2}(\frac{1}{\mathrm{D}-1}-\frac{1}{\mathrm{D}+1}){\mathrm{x}}^2\sin 3\mathrm{x}$$$$=\mathrm{A}-\mathrm{B}$$$$\mathrm{A}=\frac{1}{2(\mathrm{D}-1)}{\mathrm{x}}^2\sin 3\mathrm{x}$$$$=\frac{{\mathrm{e}}^{\mathrm{x}}}{2}\int {\mathrm{x}}^2{\mathrm{e}}^{-\mathrm{x}}\sin 3\mathrm{xdx}$$$$=\frac{1}{500}((-{25\mathrm{x}}^2+40\mathrm{x}+13)\sin 3\mathrm{x}+(-{75\mathrm{x}}^2-30\mathrm{x}+9)\cos 3\mathrm{x})$$$$\mathrm{B}=\frac{1}{2(\mathrm{D}+1)}{\mathrm{x}}^2\sin 3\mathrm{x}$$$$=\frac{{\mathrm{e}}^{-\mathrm{x}}}{2}\int {\mathrm{x}}^2{\mathrm{e}}^{\mathrm{x}}\sin 3\mathrm{xdx}$$$$=\frac{1}{500}(({25\mathrm{x}}^2+40\mathrm{x}-13)\sin 3\mathrm{x}+(-{75\mathrm{x}}^2+30\mathrm{x}+9)\cos 3\mathrm{x})$$$$\Rightarrow {\mathrm{y}}_{\mathrm{p}}=\frac{1}{250}((-{25\mathrm{x}}^2+13)\sin 3\mathrm{x}-30\mathrm{xcos}3\mathrm{x})$$$$\Rightarrow \mathrm{y}={\mathrm{C}}_1{\mathrm{e}}^{\mathrm{x}}+{\mathrm{C}}_2{\mathrm{e}}^{-\mathrm{x}}+\frac{1}{250}((-{25\mathrm{x}}^2+13)\sin 3\mathrm{x}-30\mathrm{xcos}3\mathrm{x})$$

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