The general solution of the differential equation x(y2.exy+exy)dy = y(exy−y2.exy)dx is

x(y2.exy+exy)dy=y(exy−y2.exy)dx⇒y2(xdy+ydx)exy=exy(ydx−xdy)⇒exyd(xy)=ex/yydx−xdyy2=exy.d(xy)⇒exy=exy+c

The general solution of the differential equation x(y2.exy+exy)dy = y(exy−y2.exy)dx is

x(y2.exy+exy)dy=y(exy−y2.exy)dx⇒y2(xdy+ydx)exy=exy(ydx−xdy)⇒exyd(xy)=ex/yydx−xdyy2=exy.d(xy)⇒exy=exy+c

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The general solution of the differential equation $x (y^{2} . e^{x y} + e^{\frac{x}{y}}) d y = y (e^{\frac{x}{y}} - y^{2} . e^{x y}) d x$ is

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$e^{x y} + e^{\frac{x}{y}} = c$

$e^{x y} = e^{\frac{y}{x}} + c$

$e^{x y} = e^{\frac{x}{y}} + c$

$e^{x y} + e^{\frac{y}{x}} = c$

answer is A.

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Detailed Solution

$\begin{array}{l} x (y^{2} . e^{x y} + e^{\frac{x}{y}}) d y = y (e^{\frac{x}{y}} - y^{2} . e^{x y}) d x \\ \Rightarrow y^{2} (x d y + y d x) e^{x y} = e^{\frac{x}{y}} (y d x - x d y) \\ \Rightarrow e^{x y} d (x y) = e^{x / y} \frac{y d x - x d y}{y^{2}} = e^{\frac{x}{y}} . d (\frac{x}{y}) \\ \Rightarrow e^{x y} = e^{\frac{x}{y} + c} \end{array}$