Week 3

Integrating Factorss

Given an equation of the form

\[y'+p(x)y=f(x)\]

we may define an integrating factor

\[r=e^{\int p(x)\, dx}\]

such that by multiplying each term on both sides of the equation, we derive a new equation

\[\frac{d}{dx}\left[y\cdot r(x) \right]=f(x)\cdot r(x)\]

Integrating both sides yields

\[y\cdot r(x)=\int{f(x)\cdot r(x)}\]

which may then be solved to find a solution in terms of $y$.