Method for First-Order Homogenous Differential Equations
- solves first-order differential equations that is neither separable nor linear but is homogenous
- idea is to turn a (non-linear non-separable homogenous differential equation) into a (separable one)
Method
given a first-order homogenous differential equation
- 𝑑𝑦/𝑑𝑥 = 𝐹(𝑦/𝑥)
let 𝑣 = 𝑦/𝑥, thus:
- 𝑦 = 𝑣𝑥
- (𝑑/𝑑𝑥)𝑦 = (𝑑/𝑑𝑥)𝑣𝑥
- (𝑑𝑦/𝑑𝑥) = (𝑑/𝑑𝑥)𝑣𝑥
- (𝑑𝑦/𝑑𝑥) = (𝑣𝑥)’
- (𝑑𝑦/𝑑𝑥) = 𝑣 + 𝑥·(𝑑𝑣/𝑑𝑥) # product rule
now substitute [𝑣 = 𝑦/𝑥] and [(𝑑𝑦/𝑑𝑥) = 𝑣 + 𝑥·(𝑑𝑣/𝑑𝑥)] into differential equation
- 𝑑𝑦/𝑑𝑥 = 𝐹(𝑦/𝑥)
- 𝑣 + 𝑥·(𝑑𝑣/𝑑𝑥) = 𝐹(𝑣)
- 𝑥·(𝑑𝑣/𝑑𝑥) = 𝐹(𝑣) - 𝑣
- 𝑥/𝑑𝑥 = [𝐹(𝑣) - 𝑣] / 𝑑𝑣
then integrate both sides
replace 𝑣 with 𝑦/𝑥 and then solve for 𝑦