Question

Solution of the differential equation
$\left[y\left(1+x^{-1}\right)+\sin y\right] d x+(x+\log x+x \cos y) d y=0$ is

• A. $x y+y \log x=c$
• B. $x y+x \sin y=c$
• C. $x y+y \log x+x \sin y=c$
• D. None of these

Answer 1

Answer: (C)

The given diff. equation can be written as
$\Rightarrow (y d x+x d y)+\left(\frac{y}{x} d x+\log x\right) d y + \sin y\, dx + x \cos y\, dy = 0$
$\Rightarrow d(x y)+d(y \log x)+d(x \sin y)=0$
Integrating, we get
$x y+y \log x+x \sin y=c$.