Question

The solution of the differential equation $(x^2-y{x}^2)\frac{dy}{dx}+y^2+x{y}^2=0$ is

• A. $\log \left(\frac{x}{y}\right)=\frac{1}{x}+\frac{1}{y}+C$
• B. $\log \left(\frac{y}{x}\right)=\frac{1}{x}+\frac{1}{y}+C$
• C. $\log (xy)=\frac{1}{x}+\frac{1}{y}+C$
• D. $\log (xy)+\frac{1}{x}+\frac{1}{y}=C$

Answer 1

Answer: (A)

Given differential equation is
$(x^2-y{x}^2)\frac{dy}{dx}+y^2=x{y}^2=0$
$\Rightarrow$ $\frac{1-y}{y^2}dy+\frac{1+x}{x^2}dx=0$
$\Rightarrow$ $\left(\frac{1}{y^2}-\frac{1}{y}\right)dy+\left(\frac{1}{x^2}+\frac{1}{x}\right)dx=0$
On integrating both sides, we get the required solution
$\frac{-2}{y}-\log y-\frac{2}{x}+\log x=C$
$\Rightarrow$ $\log \left(\frac{x}{y}\right)=\frac{1}{x}+\frac{1}{y}+C$