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Question Number 138240 by bobhans last updated on 11/Apr/21

$$(x-1)\frac{dy}{dx}+xy=2{xe}^{-x}$$

Answered by EDWIN88 last updated on 11/Apr/21

$$\frac{dy}{dx}+\frac{x}{x-1}y=\frac{2x}{e^x(x-1)}$$$$putIF=e^{\int \frac{x}{x-1}dx}=e^{\int \frac{x-1+1}{x-1}dx}=e^{x+\ln (x-1)}$$$$IF=e^x(x-1)$$$$\iff y=\frac{\int \frac{2x}{e^x(x-1)}.e^x(x-1)dx+C}{e^x(x-1)}$$$$\iff y=\frac{x^2+C}{e^x(x-1)}.$$

Answered by physicstutes last updated on 11/Apr/21

$$\frac{dy}{dx}+\frac{xy}{(x-1)}=\frac{2{xe}^{-x}}{(x-1)}$$$$I_{\mathrm{fac}}=e^{\int p\left(x\right)dx}=e^{\int \frac{x}{x-1}dx}=e^{\int (1+\frac{1}{x-1})}=e^{x+\ln (x-1)}=(x-1)e^x$$$$\Rightarrow (1-x)e^x\frac{dy}{dx}+\frac{(x-1){xye}^x}{(x-1)}=2x(x-1)$$$$\Rightarrow \frac{d}{dx}\left({ye}^x(x-1)\right)=2x(x-1)$$$$\Rightarrow {ye}^x(x-1)=\int (2x^2-2x)dx$$$$\Rightarrow {ye}^x(x-1)=\frac{2}{3}{x}^3-x^2+k$$$$y=\frac{2}{3}\frac{x^3}{e^x(x-1)}-\frac{x^2}{e^x(x-1)}+\frac{k}{e^x(x-1)}$$$$\mathrm{hence}y=\frac{A+x^2}{e^x(x-1)}$$

Answered by mathmax by abdo last updated on 11/Apr/21

$$(\mathrm{x}-1){\mathrm{y}}^{{}^{\prime }}+\mathrm{xy}={2\mathrm{xe}}^{-\mathrm{x}}$$$$\mathrm{h}\to (\mathrm{x}-1){\mathrm{y}}^{{}^{\prime }}+\mathrm{xy}=0\Rightarrow (1-\mathrm{x}){\mathrm{y}}^{{}^{\prime }}=\mathrm{xy}\Rightarrow \frac{{\mathrm{y}}^{{}^{\prime }}}{\mathrm{y}}=\frac{\mathrm{x}}{1-\mathrm{x}}\Rightarrow$$$$\log \mid \mathrm{y}\mid =\int \frac{\mathrm{x}}{1-\mathrm{x}}\mathrm{dx}=-\int \frac{1-\mathrm{x}-1}{1-\mathrm{x}}\mathrm{dx}=-\mathrm{x}+\int \frac{\mathrm{dx}}{1-\mathrm{x}}=-\mathrm{x}-\log \mid 1-\mathrm{x}\mid +\mathrm{c}$$$$\Rightarrow \mathrm{y}\left(\mathrm{x}\right)={\mathrm{ke}}^{-\mathrm{x}}.\frac{1}{\mid \mathrm{x}-1\mid }\mathrm{solution}\mathrm{on}\mathrm{w}=\{\mathrm{x}/\mathrm{x}>1\}\Rightarrow \mathrm{mvc}\mathrm{method}\to$$$$\mathrm{y}\left(\mathrm{x}\right)=\mathrm{k}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}\Rightarrow {\mathrm{y}}^{{}^{\prime }}\left(\mathrm{x}\right)={\mathrm{k}}^{{}^{\prime }}.\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}+\mathrm{k}\left(\frac{-{\mathrm{e}}^{-\mathrm{x}}(\mathrm{x}-1)-{\mathrm{e}}^{-\mathrm{x}}}{{(\mathrm{x}-1)}^2}\right)$$$$={\mathrm{k}}^{{}^{\prime }}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}-\mathrm{k}\frac{{\mathrm{xe}}^{-\mathrm{x}}}{{(\mathrm{x}-1)}^2}$$$$\mathrm{e}\Rightarrow {\mathrm{k}}^{{}^{\prime }}{\mathrm{e}}^{-\mathrm{x}}-\mathrm{k}\frac{{\mathrm{xe}}^{-\mathrm{x}}}{\mathrm{x}-1}+\frac{{\mathrm{xke}}^{-\mathrm{x}}}{\mathrm{x}-1}={2\mathrm{xe}}^{-\mathrm{x}}\Rightarrow {\mathrm{k}}^{{}^{\prime }}=2\mathrm{x}\Rightarrow \mathrm{k}={\mathrm{x}}^2+\lambda \Rightarrow$$$$\mathrm{y}\left(\mathrm{x}\right)=({\mathrm{x}}^2+\lambda )\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}$$

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