Integration factor method to solve linear differential equation

solve first order linear differential equation with Integration factor method

\[ \frac{dy}{dx} + p(x)y(x) = q(x) \]
let $v(x) = \int p(x) dx$
The integration factor is $I=e^{v(x)}$.
\[ \frac{d[e^{v(x)} y(x)]}{dx} = e^{v(x)}[\frac{dy}{dx} + p(x)y(x)] = e^{v(x)}q(x) \]
The solution of the differential equation is
\[ y(x) = e^{-v(x)}[ \int e^{v(x)} q(x) dx +C ]\]

\[ [I(x)y]' = I(x)q(x) \]
\[ \int_{x_0}^x [I(x)y]' dx = I(x)y(x) - I(x_0)y(x_0) = \int_{x_0}^{x} I(x)q(x) dx \]
\[ y(x) = I(x)^{-1}[I(x_0)y(x_0)+\int_{x_0}^{x} I(x)q(x) dx] \]
Reference:
http://mathworld.wolfram.com/IntegratingFactor.html
https://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-LinearDiffEqns_Stu.pdf