A) \[\frac{e}{\sqrt{2}}\]
B) \[\sqrt{3}\,e\]
C) \[\sqrt{2}\,e\]
D) \[\frac{1}{2}\sqrt{3}\,e\]
Correct Answer: B
Solution :
\[\frac{dy}{dx}=\frac{xy}{{{x}^{2}}+{{y}^{2}}},\] it is a homogenous differential equation. Put \[y=Vx\Rightarrow \frac{dy}{dx}=V+x\frac{dV}{dx}\] \[\Rightarrow \,\,\,\,\,\,\,\,V+x\frac{dV}{dx}=\frac{V}{1+{{V}^{2}}}=\int{\frac{1+{{V}^{2}}}{{{V}^{3}}}}dV=-\int{\frac{1}{x}dx}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{1}{2{{v}^{2}}}+\ln \,\,\,V=-\ln \,\,x+c\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\frac{{{x}^{2}}}{2{{y}^{2}}}=-\ln \,y+c\] \[y(1)=1\Rightarrow c=-\frac{1}{2}\] \[\therefore \] Solution is given by \[{{x}^{2}}={{y}^{2}}(1+2\,\,\ln \,\,y)\] \[\Rightarrow \,\,\,{{x}^{2}}=3{{e}^{2}}\] \[\Rightarrow \,\,\,x=\pm \sqrt{3}e\]
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