Separable:
𝑑𝑦 𝑓 (𝑥 )
=
𝑑𝑥 𝑔(𝑦)
∫ 𝑔(𝑦)𝑑𝑦 = ∫ 𝑓(𝑥)𝑑𝑥
Homogenous:
𝑦 = 𝑢𝑥
x = uy
Exact:
𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0
𝑢 = ∫ 𝑀𝑑𝑥 + 𝑘(𝑦) = ∫ 𝑁𝑑𝑦 + 𝑙(𝑥)
If NOT Exact:
𝑃(𝑥, 𝑦)𝑑𝑥 + 𝑄(𝑥, 𝑦)𝑑𝑦 = 0
1
𝑅(𝑥) = (𝑃𝑦 − 𝑄𝑥 )
𝑄
𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝐹𝑎𝑐𝑡𝑜𝑟 = 𝐹(𝑥) = 𝑒𝑥𝑝 ∫ 𝑅 (𝑥 )𝑑𝑥
1
1
𝑅 (𝑦) = − (𝑃𝑦 − 𝑄𝑥 ) = (𝑄𝑥 − 𝑃𝑦 )
𝑃
𝑃
𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝐹𝑎𝑐𝑡𝑜𝑟 = 𝐹(𝑦) = 𝑒𝑥𝑝 ∫ 𝑅 (𝑦)𝑑𝑦
Linear First Order:
𝑦 ′ + 𝑝(𝑥)𝑦 = 𝑟(𝑥)
𝑦 = 𝑒 −ℎ(𝑥) [∫ 𝑒 ℎ(𝑥) 𝑟(𝑥 )𝑑𝑥 + 𝑐],
Bernoulli:
𝑦 ′ + 𝑝(𝑥)𝑦 = 𝑔(𝑥)𝑦 𝑎 ,
ℎ(𝑥) = ∫ 𝑝(𝑥 )𝑑𝑥
𝑢 = 𝑦1−𝑎