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Question Number 152801 by SANOGO last updated on 01/Sep/21

$$resouds$$$$\mid 1-x\mid y^{\prime }+xy=x$$

Commented by mathmax by abdo last updated on 02/Sep/21

$$\mathrm{case}1\mathrm{if}\mathrm{x}>1\mathrm{e}\Rightarrow (\mathrm{x}-1){\mathrm{y}}^{,}+\mathrm{xy}=\mathrm{x}\left(\mathrm{e}\right)$$$$\left(\mathrm{he}\right)\to (\mathrm{x}-1){\mathrm{y}}^{{}^{\prime }}+\mathrm{xy}=0\Rightarrow (\mathrm{x}-1){\mathrm{y}}^{{}^{\prime }}=-\mathrm{xy}\Rightarrow \frac{{\mathrm{y}}^{{}^{\prime }}}{\mathrm{y}}=-\frac{\mathrm{x}}{\mathrm{x}-1}$$$$=-\frac{\mathrm{x}-1+1}{\mathrm{x}-1}=-1-\frac{1}{\mathrm{x}-1}\Rightarrow \ln \mid \mathrm{y}\mid =-\mathrm{x}-\ln (\mathrm{x}-1)\Rightarrow$$$$\mathrm{y}={\mathrm{e}}^{-\mathrm{x}}\times \frac{1}{\mathrm{x}-1}=\mathrm{K}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}$$$$\mathrm{mvc}\mathrm{method}{\mathrm{y}}^{{}^{\prime }}={\mathrm{K}}^{{}^{\prime }}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}+\mathrm{K}\times \frac{-{\mathrm{e}}^{-\mathrm{x}}(\mathrm{x}-1)-{\mathrm{e}}^{-\mathrm{x}}}{{(\mathrm{x}-1)}^2}$$$$={\mathrm{K}}^{{}^{\prime }}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}-\mathrm{K}\frac{{\mathrm{xe}}^{-\mathrm{x}}}{{(\mathrm{x}-1)}^2}$$$$\left(\mathrm{e}\right)\Rightarrow {\mathrm{K}}^{{}^{\prime }}{\mathrm{e}}^{-\mathrm{x}}-\mathrm{K}\frac{{\mathrm{xe}}^{-\mathrm{x}}}{\mathrm{x}-1}+\mathrm{Kx}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}=\mathrm{x}\Rightarrow {\mathrm{K}}^{{}^{\prime }}={\mathrm{xe}}^{\mathrm{x}}\Rightarrow$$$$\mathrm{K}=\int {\mathrm{xe}}^{\mathrm{x}}\mathrm{dx}={\mathrm{xe}}^{\mathrm{x}}-\int {\mathrm{e}}^{\mathrm{x}}\mathrm{dx}={\mathrm{xe}}^{\mathrm{x}}-{\mathrm{e}}^{\mathrm{x}}+\lambda =(\mathrm{x}-1){\mathrm{e}}^{\mathrm{x}}+\lambda \Rightarrow$$$$\mathrm{the}\mathrm{general}\mathrm{so}<\mathrm{ution}\mathrm{is}$$$$\mathrm{y}=\mathrm{K}\left(\mathrm{x}\right)\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}=\{(\mathrm{x}-1){\mathrm{e}}^{\mathrm{x}}+\lambda \}\frac{{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}=1+\frac{\lambda {\mathrm{e}}^{-\mathrm{x}}}{\mathrm{x}-1}$$$$\mathrm{case}2\mathrm{x}<1\Rightarrow (1-\mathrm{x}){\mathrm{y}}^{{}^{\prime }}+\mathrm{xy}=\mathrm{x}\mathrm{we}\mathrm{follow}\mathrm{the}\mathrm{same}\mathrm{way}....$$$$$$

Commented by SANOGO last updated on 02/Sep/21

$$mercibienledoyen$$

Commented by Mathspace last updated on 02/Sep/21

$$thankyousir$$

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