Solving xy' + 2y = sin(x) - Integrator Factor

[paula17]

Hi guys, i need your help. I am not sure what is the integrator factor in this diferential equation to start solving it.

xy' + 2y = sin(x)

Thanks a lot

 

[LeBrad]

You need to first write it as
y' + p(x)y = q(x)

Then evaluate
$$\mu = exp(\int p(x) dx)$$

Is that what you mean by integrating factor?

 

[HallsofIvy]

Or, if you would prefer to derive it yourself, an integrating factor is a function p(x) such that multiplying the equation by it, the left side becomes an "exact derivative":
$$\frac{d(p(x)xy)}{dx}= p(x)x\frac{dy}{dx}+ 2p(x)y$$
Since
[tex]\frac{d(p(x)xy)}{dx}= p(x)x\frac{dy}{dx}+ p(x)y+ p'(x)xy[/itex] <br /> that means we must have xp'+ p= 2p or xp'= p, a simple separable differential equation for p. Solving it you get exactly what LeBrad said.[/tex]