Question

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

• A. $\log \frac{x}{ky}=\frac{x+y}{xy}$
• B. $\log \frac{x}{ky}=\frac{x-y}{xy}$
• C. $\log \frac{x}{ky}=\frac{x-y}{y}$
• D. $\log \frac{x}{ky}=\frac{x+y}{y}$

Answer 1

The correct option is A $\log \frac{x}{ky}=\frac{x+y}{xy}$

$x^2(1-y)\frac{dy}{dx}+y^2(1+x)=0$
or $\frac{1-y}{y^2}dy+\frac{1+x}{x^2}dx=0.$
or $(\frac{1}{y^2}-\frac{1}{y})dy+(\frac{1}{x^2}+\frac{1}{x})dx=0.$
Integrating, we get
$\log x-\log y-(\frac{1}{x}+\frac{1}{y})=c=\log k$ (say)
or $\log \frac{x}{ky}=\frac{x+y}{xy}$