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 \[\frac{dy}{dx}=\frac{{{y}^{2}}}{xy-{{x}^{2}}}\] It is a homogeneous differential equation Put\[y=vx\] \[\Rightarrow \] \[\frac{dy}{dx}=v+x\frac{dv}{dx}\] \[\therefore \] \[v+x\frac{dv}{dx}=\frac{{{v}^{2}}{{x}^{2}}}{x\cdot vx-{{x}^{2}}}\] \[\Rightarrow \] \[v+x\frac{dv}{dx}=\frac{{{v}^{2}}}{v-1}\] \[\Rightarrow \] \[x\frac{dv}{dx}=\frac{v}{v-1}\] \[\Rightarrow \] \[\left( 1-\frac{1}{v} \right)dv=\frac{dx}{x}\] On integrating, we get \[v-\log v=\log x-\log c\] \[\Rightarrow \] \[\frac{y}{x}=\log \frac{y}{x}\cdot x\cdot \frac{1}{c}\] \[\Rightarrow \] \[y=c{{e}^{y/x}}\]
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