Question

The solution of the differential equation $\frac{ydx - xdy}{xy}=xdx+ydy$ is (where, $C$ is an arbitrary constant)

• A. $\frac{x}{y}=x+y+C$
• B. $\frac{x}{y}=\frac{x^{2} + y^{2}}{2}+C$
• C. $ln \left(\frac{x}{y}\right)=x^{2}+y^{2}+C$
• D. $2ln\left(\frac{x}{y}\right)=x^{2}+y^{2}+C$

Answer 1

Answer: (D)

The given equation is $\frac{1}{\left(\frac{x}{y}\right)}\cdot \frac{ydx - xdy}{y^{2}}=xdx+ydy$
or $d \left(\ln \left(\frac{ x }{ y }\right)\right)= x d x + ydy$
On integrating, we get,
$\int d \left(\ln \left(\frac{ x }{ y }\right)\right)=\int xdx +\int ydy$
$\Rightarrow \ln \left(\frac{ x }{ y }\right)=\frac{ x ^{2}}{2}+\frac{ y ^{2}}{2}+ k$
$\Rightarrow 2 \ln \left(\frac{ x }{ y }\right)= x ^{2}+ y ^{2}+ C$