A) \[\frac{1}{2}+\frac{1}{\sqrt{e}}\]
B) \[\frac{3}{2}-\sqrt{e}\]
C) \[\frac{5}{2}+\frac{1}{\sqrt{e}}\]
D) \[\frac{3}{2}-\frac{1}{\sqrt{e}}\]
Correct Answer: D
Solution :
\[{{y}^{2}}dx+xdy=\frac{dy}{y}\] \[\frac{dx}{dy}+\frac{x}{{{y}^{2}}}=\frac{1}{{{y}^{3}}}\] \[IF={{e}^{\int_{{}}^{{}}{\frac{1}{{{y}^{2}}}dy}}}={{e}^{-\frac{1}{y}}}\] \[{{e}^{-\frac{1}{y}}}.x=\int_{{}}^{{}}{{{e}^{-\frac{1}{y}}}}.\frac{1}{{{y}^{3}}}dy+C\] \[x{{e}^{-\frac{1}{y}}}={{e}^{-\frac{1}{y}}}+\frac{{{e}^{-\frac{1}{y}}}}{y}+C\] \[C=-\frac{1}{e}\] \[x=\frac{3}{2}-\frac{1}{\sqrt{e}}\]when\[y=2\]
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