$Solution of the differential equation \frac{\sqrt{x} d x + \sqrt{y} d y}{\sqrt{x} d x - \sqrt{y} d y} = \sqrt{\frac{y^{3}}{x^{3}}} is given by (Base of all \log = e)$

Moderate

A

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

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

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

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

Solution

$We have \frac{\sqrt{x} d x + \sqrt{y} d y}{\sqrt{x} d x - \sqrt{y} d y} = \sqrt{\frac{y^{3}}{x^{3}}}$
$\begin{array}{l} \Rightarrow \frac{d (x^{\frac{3}{2}}) + d (y^{\frac{3}{2}})}{d (x^{\frac{3}{2}}) - d (y^{\frac{3}{2}})} = \frac{y^{\frac{3}{2}}}{x^{\frac{3}{2}}} \\ \Rightarrow \frac{d u + d v}{d u - d v} = \frac{v}{u} \\ (Where u = x^{\frac{3}{2}} and v = y^{\frac{3}{2}}) \\ \Rightarrow u d u + u d v = v d u - v d v \\ \Rightarrow u d u + v d v = v d u - u d v \\ \Rightarrow \frac{u d u + v d v}{u^{2} + v^{2}} = \frac{v d u - u d v}{u^{2} + v^{2}} \end{array}$
$\begin{array}{l} \Rightarrow \frac{d (u^{2} + v^{2})}{u^{2} + v^{2}} = - 2 {dtan}^{- 1} (\frac{v}{u}) \\ On integrating we get \\ \log (u^{2} + v^{2}) = - 2 \tan^{- 1} (\frac{v}{u}) - 2 c \\ \Rightarrow \frac{1}{2} \log (x^{3} + y^{3}) + \tan^{- 1} {(\frac{y}{x})}^{\frac{3}{2}} + c = 0 \end{array}$

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