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$\large\frac{dy}{dx} =\large\frac{y+x+1}{y+x +5}$

$(a)\;y+x^2-1+\log (y+x+5)=x^2+c \\ (b)\;y+x+1+ \log (y+x+5)=x^2+c \\ (c)\;y+x+5+\log (y+x+1)=x^2+c \\ (d)\;y+x+2\log (y+x+3)=x^2+c $
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$y+x+1=t$
$\large\frac{dy}{dx}$$+1=\large\frac{dt}{dx}$
$\large\frac{dt}{dx}$$-1=\large\frac{t}{t+4}$
$\large\frac{dt}{dx}=\large\frac{t}{t+4}+1$
$\large\frac{2t+4}{t+4}$
$\large\frac{t+4}{2t+4 }$$dt=dx$
$\int \large\frac{t+2+2}{t+2}$$dt=dx$
$\int \bigg( \large\frac{t+2}{t+2} +\frac{2}{t+2}\bigg)$$dt=2x$
$t+ 2 \log (t+2)=x^2+c$
$y+x+1+2 \log (y+x+3)=x^2+c$
$ y+x+2\log (y+x+3)=x^2+c $
Hence d is the correct answer.
answered Feb 3, 2014 by meena.p
 

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