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Question: Solution of the differential equation \(\left\{ \frac{1}{x} - \frac{y^{2}}{(x - y)^{2}} \right\}\)d...

Solution of the differential equation

{1x−y2(x−y)2}\left\{ \frac{1}{x} - \frac{y^{2}}{(x - y)^{2}} \right\}dx + {x2(x−y)2−1y}\left\{ \frac{x^{2}}{(x - y)^{2}} - \frac{1}{y} \right\}dy = 0 is

A

ln∣xy∣\left| \frac{x}{y} \right| + xyx−y\frac{xy}{x - y} = c

B

xyx−y\frac{xy}{x - y} = cex/y

C

ln |xy| = c + xyx−y\frac{xy}{x - y}

D

None of these

Answer

ln∣xy∣\left| \frac{x}{y} \right| + xyx−y\frac{xy}{x - y} = c

Explanation

Solution

(dxx−dyy)+(x2dy−y2dx(x−y)2)\left( \frac{dx}{x} - \frac{dy}{y} \right) + \left( \frac{x^{2}dy - y^{2}dx}{(x - y)^{2}} \right) = 0

(dxx−dyy)\left( \frac{dx}{x} - \frac{dy}{y} \right)+ (dyy2−dxx2)(1y−1x)2\frac{\left( \frac{dy}{y^{2}} - \frac{dx}{x^{2}} \right)}{\left( \frac{1}{y} - \frac{1}{x} \right)^{2}} = 0

(dxx−dyy)\left( \frac{dx}{x} - \frac{dy}{y} \right)+ [dyy2−dxx2(1x−1y)2]\left\lbrack \frac{\frac{dy}{y^{2}} - \frac{dx}{x^{2}}}{\left( \frac{1}{x} - \frac{1}{y} \right)^{2}} \right\rbrack = 0

ln |x| – ln |y| – 1(1x−1y)\frac{1}{\left( \frac{1}{x} - \frac{1}{y} \right)} = c

ln ∣xy∣\left| \frac{x}{y} \right| + xyx−y\frac{xy}{x - y} = c