Question

The general solution of $y^{2}dx+(x^2-xy+y^2)dy=0$ is

• A. $ta{n}^{-1}\left(\frac{x}{y}\right)+logy+c=0$
  
• B. $2ta{n}^{-1}\left(\frac{x}{y}\right)+logx+c=0$
  
• C. $log(y+\sqrt{x^2+y^2})+logy+c=0$
  
• D. $sin{h}^{-1}\left(\frac{x}{y}\right)+logy+c=0$

Answer 1

The correct option is A $ta{n}^{-1}\left(\frac{x}{y}\right)+logy+c=0$


$\frac{dx}{dy}+\frac{x^2-xy+y^2}{y^2}=0\frac{dx}{dy}+{\left(\frac{x}{y}\right)}^2-\left(\frac{x}{y}\right)+1=0$
Put $v=\frac{x}{y}\Rightarrow x=vy\Rightarrow \frac{dx}{dy}=v+y\frac{dv}{dy}$
$v+y\frac{dv}{dy}+v^2-v+1=0\Rightarrow \frac{dv}{v^2+1}+\frac{dy}{y}=0\Rightarrow \int \frac{dv}{v^2+1}+\int \frac{dy}{y}=0\Rightarrow ta{n}^{-1}\left(v\right)+logy+C=0\Rightarrow ta{n}^{-1}\left(\frac{x}{y}\right)+logy+c=0$.