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Solution of differential equation : $y(2xy+e^x)dx=e^xdy$

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

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$y(2xy+e^x)dx=e^{x}dy$
$(y(e^x dx-e^{x}dy)+2xy^2dx=0$
Divide by $y^2$
$\large\frac{e^x}{y}$$ dx-\large\frac{e^x}{y^2}$$dy+2xdx=0$
$\large\frac{ye^{x}dx-e^x dy}{y^2}$$+d(x^2)=0$
$d \bigg(\large\frac{e^x}{y} \bigg)$$+d(x^2)=0$
$\large\frac{e^x}{y}$$+x^2=c$
Hence d is the correct answer.
answered Feb 4, 2014 by meena.p
 

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