Regarding linear differential equations, why is the integrating factor $$e ^{\int\P(x)}$$, where did that come from?

Question

anonymous · Answer

It comes from the product rule.

myininaya · Answer

$$y'+p(x)y=q(x)$$
we need to multiply both sides by v(x) such that v*p=v' so that we can write
vy'+vpy=vq
remember we chose a v such that vp=v'
vy'+v'y=vq
(yv)'=vq
and so on...
any ways what is v
well v'=dv/dx
so $$\frac{dv}{dx}=vp$$
we can use separation of variables to find v
$$\frac{1}{v} dv=p dx$$
integrate both sides
$$lnv=\int\limits_{}^{}pdx+C$$
$$v=e^{\int\limits_{}^{}p d}+C$$

myininaya · Answer

v>0

myininaya · Answer

any questions?

myininaya · Answer

by the way we don't need the constant
we can take the constant to be zero

myininaya · Answer

but it doesn't matter you can always divide the constant out in the equation if you do side to multiply by it

anonymous · Answer

Excellent, Thank you.