A) \[y=c{{e}^{x/y}}\]
B) \[y=c{{e}^{y/x}}+x\]
C) \[y=c{{e}^{y/x}}\]
D) \[xy=c{{e}^{y/x}}\]
Correct Answer: C
Solution :
Given, equation is \[\frac{dy}{dx}=\frac{{{y}^{2}}}{xy-{{x}^{2}}}\] which is a homogeneous differential equation. \[\therefore \]Put\[y=vx,\] \[\Rightarrow \] \[\frac{dy}{dx}=v+x\frac{dv}{dx}\] \[v+x\frac{dv}{dx}=\frac{{{v}^{2}}{{x}^{2}}}{x.vx-{{x}^{2}}}\] \[\Rightarrow \] \[v+x\frac{dv}{dx}=\frac{{{v}^{2}}}{v-1}\] \[\Rightarrow \] \[x\frac{dv}{dx}=\frac{v}{v-1}\] \[\Rightarrow \] \[\frac{v-1}{v}dv=\frac{dx}{x}\] On integrating, we get \[v-log\text{ }v=log\text{ }x-log\text{ c}\] \[\Rightarrow \] \[\frac{y}{x}=\log \frac{y}{x}.x.\frac{1}{c}\] \[\Rightarrow \] \[y=c{{e}^{y/x}}\]
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