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Question Number 169287 by cortano1 last updated on 28/Apr/22

$$solve:x^3\frac{d^{3}y}{{dx}^3}+2x^2\frac{d^{2}y}{{dx}^2}+2y=10(x+\frac{1}{x})$$

Commented by infinityaction last updated on 01/May/22

$$letz=\log {}_{\mathit{e}}x\Rightarrow x=e^{\mathit{z}}$$$$\{D(D-1)(D-2)+2D(D-1)+2\}y=10(e^{\mathit{z}}+e^{-z})$$$$where\frac{d}{dz}=D$$$$A\mathrm{uxiliary}\mathrm{Equation}$$$$m^3-m^2+2=0$$$$(m+1)(m^2-2m+2)=0$$$$m=-1,1\underset{-}{+}i$$$$\mathit{C}.\mathit{F}={\mathit{c}}_1{\mathit{e}}^{-\mathit{x}}+{\mathit{e}}^{\mathit{z}}({\mathit{c}}_2\mathbf{cos}\mathit{z}+{\mathit{c}}_3\mathbf{sin}\mathit{z})$$$$\mathit{C}.\mathit{F}={\mathit{c}}_1{\mathit{x}}^{-1}+\mathit{x}\{{\mathit{c}}_2\mathbf{cos}\left(\mathbf{log}\mathit{x}\right)+{\mathit{c}}_3\mathbf{sin}\left(\mathbf{log}\mathit{x}\right)\}$$$$\mathit{P}.\mathit{I}=\frac{1}{(\mathit{D}+1)({\mathit{D}}^2-2\mathit{D}+2)}10({\mathit{e}}^{\mathit{z}}+{\mathit{e}}^{-\mathit{z}})$$$$\mathit{P}.\mathit{I}=10\{\frac{1}{(\mathit{D}+1)({\mathit{D}}^2-2\mathit{D}+2)}{\mathit{e}}^{\mathit{z}}+\frac{1}{(\mathit{D}+1)({\mathit{D}}^2-2\mathit{D}+2)}{\mathit{e}}^{-\mathit{z}}\}$$$$P.I=10\{\frac{e^{\mathit{z}}}{2}+\frac{1}{D+1}\cdot \frac{e^{-\mathit{z}}}{1+2+2}\}$$$$P.I=10\{\frac{e^z}{2}+\frac{1}{5}\frac{1}{(D+1)}{e}^{-z}\}$$$$P.I=10\{\frac{e^{-\mathit{z}}}{2}+\frac{e^{-\mathit{z}}}{5}\frac{1}{(D-1+1)}\cdot 1\}$$$$P.I=10\{\frac{e^{\mathit{z}}}{2}+\frac{{ze}^{-\mathit{z}}}{5}\}=5x+2x^{-1}\log x$$$$\mathit{y}={\mathit{c}}_1{\mathit{x}}^{-1}+\mathit{x}\{{\mathit{c}}_2\mathbf{cos}\left(\mathbf{log}\mathit{x}\right)+{\mathit{c}}_3\mathbf{sin}\left(\mathbf{log}\mathit{x}\right)\}+5\mathit{x}+2{\mathit{x}}^{-1}\mathbf{log}\mathit{x}$$$$$$$$$$

Commented by cortano1 last updated on 28/Apr/22

$$nice$$

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