Question

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

Answer 1

It comes from the product rule.

Answer 2

\[y'+p(x)y=q(x)\]
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
\[lnv=\int\limits_{}^{}pdx+C\]
\[v=e^{\int\limits_{}^{}p d}+C\]

Answer 3

v>0

Answer 4

any questions?

Answer 5

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

Answer 6

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

Answer 7

Excellent, Thank you.