integrating factors

Question

How do I use integrating factors solve the following equation?

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

Answer 1

Given

x ( dy / dx ) + y = x sinx

Divide both sides with x

Now compare it with the first order differential equation

 Integrating factor I for first order differential equation

    

Integrating factor for given differential equation with p = 1 / x

Multiply the differential equation (1)  by I =x , we get 

x ( dy / dx ) + y = x sinx

y = x sinx - x ( dy / dx )

Multiply both sides by dx

y dx  = x sinx dx - x ( dy / dx ) * dx

y dx  = x sinx dx - x dy

Integrate both sides now  

 ʃ y dx  =  ʃ x sinx dx -  ʃ x dy

 y x = - xcosx + sin x - xy

y x + x y  = - x cosx +  sin x

2 x y  = - x cosx +  sin x

y  = - xcosx  / 2x + ( sin x / 2 x )