xdx+ydyxdx−ydy=y3x3 The solution of this differential equation is

xdx+ydyxdx+ydy=y3y3d(x3/2)+d(y3/2)d(x3/2)−d(y3/2)=y3/2x3/2du+dvdu−dv=vu(u=x3/2)(v=y3/2)udv+vdv=vdu−vduudv+vduv2+u2=vdu−udvu2+v2d(u2+v2)u2+v2=−2d(tan−1uv)an integr then we getlog(u2+v2)=−2tan−1(uv)+c12h(x3+y3)+tan−1(xy)3/2=c2=c

xdx+ydyxdx−ydy=y3x3 The solution of this differential equation is

xdx+ydyxdx+ydy=y3y3d(x3/2)+d(y3/2)d(x3/2)−d(y3/2)=y3/2x3/2du+dvdu−dv=vu(u=x3/2)(v=y3/2)udv+vdv=vdu−vduudv+vduv2+u2=vdu−udvu2+v2d(u2+v2)u2+v2=−2d(tan−1uv)an integr then we getlog(u2+v2)=−2tan−1(uv)+c12h(x3+y3)+tan−1(xy)3/2=c2=c

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$\frac{\sqrt{x} d x + \sqrt{y} d y}{\sqrt{x} d x - \sqrt{y} d y} = \sqrt{\frac{y^{3}}{x^{3}}}$ The solution of this differential equation is

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$\frac{3}{2} \log (\frac{y}{x}) + \log | \frac{x^{\frac{3}{2}} + y^{\frac{3}{2}}}{x^{\frac{3}{2}}} | + \tan^{- 1} {(\frac{y}{x})}^{\frac{3}{2}} + c = 0$

$\frac{2}{3} \log (\frac{y}{x}) + \log | \frac{x^{\frac{3}{2}} + y^{\frac{3}{2}}}{x^{\frac{3}{2}}} | + \tan^{- 1} {(\frac{y}{x})}^{\frac{3}{2}} + c = 0$

$\frac{2}{3} (\frac{y}{x}) + \log (\frac{x + y}{x}) + \tan^{- 1} (\frac{y^{\frac{3}{2}}}{x^{\frac{3}{2}}}) + c = 0$

$\frac{1}{2} \log | x^{3} + y^{3} | + \tan^{- 1} (\frac{y^{\frac{3}{2}}}{x^{\frac{3}{2}}}) + c = 0$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{\sqrt{x} d x + \sqrt{y} d y}{\sqrt{x} d x + \sqrt{y} d y} = \sqrt{\frac{y^{3}}{y^{3}}} \\ \frac{d (x^{3 / 2}) + d (y^{3 / 2})}{d (x^{3 / 2}) - d (y^{3 / 2})} = \frac{y^{3 / 2}}{x^{3 / 2}} \\ \frac{d u + d v}{d u - d v} = \frac{v}{u} (u = x^{3 / 2}) (v = y^{3 / 2}) \\ u d v + v d v = v d u - v d u \\ \frac{u d v + v d u}{v^{2} + u^{2}} = \frac{v d u - u d v}{u^{2} + v^{2}} \\ \frac{d (u^{2} + v^{2})}{u^{2} + v^{2}} = - 2 d (\tan^{- 1} \frac{u}{v}) \\ a n int e g r t h e n w e g e t \\ \log (u^{2} + v^{2}) = - 2 \tan^{- 1} (\frac{u}{v}) + c \\ \frac{1}{2} h (x^{3} + y^{3}) + \tan^{- 1} {(\frac{x}{y})}^{3 / 2} = \frac{c}{2} = c \end{array}$