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Find the solution of differential equation : $\large\frac{dy}{dx}=\frac{x^2+y^2}{xy}$

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

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$\large\frac{dy}{dx}=\large\frac{x^2+y^2}{xy}$
Homogeneous,
$\large\frac{dy}{dx} =\large\frac{1+ \bigg(\Large\frac{y}{x}\bigg)^2}{\bigg(\Large\frac{y}{x}\bigg)}$
$\large\frac{y}{x} $$=t$
$\large\frac{dy}{dx}= \large\frac{x dt}{dx}$$+t$
$\qquad= \large\frac{1+t^2}{t}$
$\large\frac{x dt}{dx}=\large\frac{1+t^2}{t} -t$
$\large\frac{x dt}{dx} = \large\frac{1+t^2-t^2}{t}$
$\int t dt=\int \large\frac{dx}{x} $
$\large\frac{t^2}{2} $$=\log x +c$
$\bigg(\large\frac{y}{x}\bigg)^2$$ =2 \log x +c$
Hence b is the correct answer.
answered Feb 3, 2014 by meena.p
 

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