Question

For the differential equation $\sec x\frac{dy}{dx}=y+\sin x$. The solution is $y=-(\sin x+A)+c{e}^{\sin x}.$ Find A

Answer 1

$\sec x\frac{dy}{dx}=y+\sin x$ $\Rightarrow \frac{dy}{dx}-y\cos x=\sin x\cos x$ ...(1)
Here $P=\cos x\Rightarrow \int Pdx=\int \cos xdx=-\sin x$
$\therefore I.F=e^{-\sin x}$
Multiplying (1) by $I.F,$ we get
$e^{-\sin x}\frac{dy}{dx}-y\cos x{e}^{-\sin x}=\sin x\cos x{e}^{-\sin x}$
Integrating both sides we get
$e^{-\sin x}y=e^{-\sin x}(-\sin x-1)+c$
$\Rightarrow y=-\left(\sin x+1\right)+c{e}^{\sin x}$