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Solve : $2e^{\Large\frac{x}{y}}(dx-dy)+dx-2e^{\Large\frac{x}{y}} \large\frac{x}{y}$$dy=0$

$(a)\;x+2ye^{\frac{x}{y}}=c \\ (b)\;2x+ye^{\frac{x}{y}}=c \\ (c)\;2(x+ye^{\frac{x}{y}})=c \\ (d)\;2x+3y e^{\frac{x}{y}}=c $

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$(1+2e^{x/y})dx+2e^{x/y}(1-x/y)dy=0$
$x=vy;dx=vdy+ydv$
$(1+2e^{v})( vdy+ydv)+2e^{v} (1-v)dy=0$
=> $ (v+2e^v)dy+y(1+2e^v)dv=0$
$\large\frac{dy}{y} +\frac{1+2e^v}{v+2e^v}$$dv=0$
Integrating $\log y +\log (v+2e^v)=\log c$
$x+2ye ^{\large\frac{x}{y} }=c$
Hence a is the correct answer
answered Feb 7, 2014 by meena.p
 

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