Question

The solution of differential equation
$\frac{dy}{dx}=-\left(\frac{y+\sin x}{x}\right)$ satisfying condition $y\left(0\right)=1$, is

• A. $\cos x=xy-1$
• B. $\cos x=xy+1$
• C. $\cos x=xy$
• D. $\cos x=x+1$

Answer 1

Answer: (B)

$\cos x=xy+1$

$xdy=-ydx-\sin xdx$
$\Rightarrow -\sin xdx=xdy+ydx$
$\Rightarrow -\sin xdx=d\left(xy\right)$
$\Rightarrow \int -\sin xdx=\int d\left(xy\right)$
$\Rightarrow \cos x=xy+c$
$y\left(0\right)=1\Rightarrow c=1$
$\Rightarrow \cos x=xy+1$