Question

The solution of the differential equation $\frac{y d x - x d y}{x y}=xdx+ydy$ 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 $\frac{1}{(\frac{x}{y})}\cdot \frac{y d x - x d y}{y^{2}}=xdx+ydy$
or $d(ln (\frac{x}{y} ))=xdx+ydy$
On integrating, we get,
$\displaystyle \int d (ln (\frac{x}{y})=\displaystyle \int x d x +\displaystyle \int y d y$
$\Rightarrow ln (\frac{x}{y})=\frac{x^{2}}{2}+\frac{y^{2}}{2}+k$
$\Rightarrow 2ln (\frac{x}{y} )=x^{2}+y^{2}+C$