Question

The general solution of the differential equation \(xsinx\frac{dy}{dx}+(xcosx+sinx)y = sinx\) is

• A. \(xsinx+ycosx = c\)
• B. \(ysinx+xcosx = c\)
• C. \(xysinx-cosx = c\)
• D. \(xysinx+cosx = c\)

Hint:

Write in the form dy/dx + Py = Q and find the integrating factor.

Answer 1

The Correct Option is D

Step 1: Spot the exact derivative:
The left side of the given equation is $x\sin x\,y' + (x\cos x + \sin x)y$. Since $\frac{d}{dx}(x\sin x) = x\cos x + \sin x$, this is $\frac{d}{dx}(xy\sin x)$.

Step 2: Integrate:
So $\frac{d}{dx}(xy\sin x) = \sin x$ and $xy\sin x = -\cos x + c$, i.e. $xy\sin x + \cos x = c$.

Final Answer:
The solution is option (D). \[ \boxed{xy\sin x+\cos x=c} \]