integrating factors

Question

How do I use integrating factors solve the following equation?

x(dy/dx)+y=x sinx

Answer 1

x(dy/dx)+y=xsin(x),

dy/dx+(y/x)=sin(x).

Integrating factor is e^∫dx/x=e^ln(x)=x.

So the integrating factor x simply returns the DE to its original form!

That means x(dy/dx)+y=d(xy)/dx.

So xy=∫xsin(x)dx.

Integrate by parts u=x, du=dx, dv=sin(x)dx, v=-cos(x).

xy=-xcos(x)+∫cos(x)dx=-xcos(x)+sin(x)+C.

y=-cos(x)+(sin(x)+C)/x. C is a constant of integration.

The DE appears to use integration by parts rather than an integrating factor.