Question

The solution of the differential equation $\left[\right(x+1)\frac{y}{x}+\sin y]dx+[x+\ln x+x\cos y]dy=0$is
(where $c$ is integration constant)

• A. $xy+x\ln y+x\sin y=c$
• B. $xy+y\ln x+x\cos y=c$
• C. $xy+y\ln x+x\sin y=c$
• D. $xy+x\ln y+x\cos y=c$

Answer 1

The correct option is C $xy+y\ln x+x\sin y=c$

We can re-write the differential equation as:
$(ydx+xdy)+[\frac{y}{x}dx+\ln xdy]+(\sin ydx+x\cos ydy)=0$
$\Rightarrow d\left(xy\right)+d\left(y\ln x\right)+d\left(x\sin y\right)=0$
By integrating both sides we get,
$\Rightarrow xy+y\ln x+x\sin y=c$