Method of Integrating Factor
- is used to solve a first-order differential equation that CANNOT be written in a Separable DE form
- though it can work on:
- first-order separable differential equations
- first-order constant coefficient differential equations
Method of Integrating Factor
given a first-order differential equation
Indent
𝑦’ + 𝑃(𝑥)𝑦 = 𝑓(𝑥)
we want to find an equation 𝑦 or set of equations that satisfies the differential equation above
Indent
𝑦 = [ ∫[𝜇(𝑥)𝑓(𝑥)]𝑑𝑥 ] / 𝜇(𝑥)
where:
Indent
𝜇(𝑥) = 𝑒∫𝑃(𝑥)𝑑𝑥
derivation:
Click here to expand...
we choose an integrating factor
Indent
𝜇(𝑥)
we multiply the integrating factor with the differential equation
𝜇(𝑥)𝑦' + 𝜇(𝑥)𝑃(𝑥)𝑦 = 𝜇(𝑥)𝑓(𝑥)
let us recall the product rule
𝑢'𝑦 + 𝑢𝑦'
(𝑢𝑦)’ =
which then we can say
Indent
∫𝑢’𝑦 + 𝑢𝑦’ = ∫(𝑢𝑦)’ = 𝑢𝑦
we need to find a 𝜇(𝑥) such that 𝜇(𝑥)𝑦’ + 𝜇(𝑥)𝑃(𝑥)𝑦 would have the same form as 𝑢𝑦’ + 𝑢’𝑦. So we want
𝑢 = 𝜇(𝑥) 𝑢' = 𝜇(𝑥)𝑃(𝑥)
we know that
𝑢/𝑑𝑥 = 𝑢'
𝑑
thus
𝜇(𝑥)/𝑑𝑥 = 𝜇(𝑥)𝑃(𝑥)
𝑑
this is separable, giving us:
𝜇(𝑥)/𝜇(𝑥) = 𝑃(𝑥)𝑑𝑥
𝑑
then we could integrate both sides:
- ∫𝑑𝜇(𝑥)/𝜇(𝑥) = ∫𝑃(𝑥)𝑑𝑥
- 𝑙𝑛[𝜇(𝑥)] = ∫𝑃(𝑥)𝑑𝑥
- 𝑒𝑙𝑛[𝜇(𝑥)] = 𝑒∫𝑃(𝑥)𝑑𝑥
- 𝜇(𝑥) = 𝑒∫𝑃(𝑥)𝑑𝑥
if we choose 𝜇(𝑥) to be of form 𝑒∫𝑃(𝑥)𝑑𝑥 then ∫[𝜇(𝑥)𝑦’ + 𝜇(𝑥)𝑃(𝑥)𝑦]𝑑𝑥 is equal to 𝜇(𝑥)𝑦. Thus:
- 𝜇(𝑥)𝑦’ + 𝜇(𝑥)𝑃(𝑥)𝑦 = 𝜇(𝑥)𝑓(𝑥)
- ∫[𝜇(𝑥)𝑦’ + 𝜇(𝑥)𝑃(𝑥)𝑦]𝑑𝑥 = ∫[𝜇(𝑥)𝑓(𝑥)]𝑑𝑥
- 𝜇(𝑥)𝑦 = ∫[𝜇(𝑥)𝑓(𝑥)]𝑑𝑥
- 𝑦 = [∫[𝜇(𝑥)𝑓(𝑥)]𝑑𝑥] / 𝜇(𝑥)
Thus:
- 𝑦 = [∫[𝜇(𝑥)𝑓(𝑥)]𝑑𝑥] / 𝜇(𝑥)
where:
- 𝜇(𝑥) = 𝑒∫𝑃(𝑥)𝑑𝑥