Bernouli differential equations
Practice problems

First order ODEs

Bernouli differential equations

A Bernouli ODE is a first-order, non-linear ODE with the standard form

𝑑 𝑦 𝑑 𝑥 + 𝑓 (𝑥) 𝑦 = 𝑔 (𝑥) 𝑦 𝑛

which in the case of $𝑛 = 0$ is a First-order linear differential equation; and for $𝑛 = 1$ is a separable differential equation. In all other cases, the ODE may be made linear by using a $𝑤$ -substitution:

𝑤 = 𝑦 1 - 𝑛 ⟹ 𝑑 𝑤 𝑑 𝑥 = (1 - 𝑛) 𝑦 - 𝑛 𝑑 𝑦 𝑑 𝑥 ⟹ 𝑑 𝑦 𝑑 𝑥 = (1 - 𝑛) - 1 𝑦 𝑛 𝑑 𝑤 𝑑 𝑥

which when entered into the original ODE gives

(1 - 𝑛) - 1 𝑦 𝑛 𝑑 𝑤 𝑑 𝑥 + 𝑓 (𝑥) 𝑦 = 𝑔 (𝑥) 𝑦 𝑛 𝑑 𝑤 𝑑 𝑥 + (1 - 𝑛) 𝑓 (𝑥) 𝑦 1 - 𝑛 = (1 - 𝑛) 𝑔 (𝑥) 𝑑 𝑤 𝑑 𝑥 + (1 - 𝑛) 𝑓 (𝑥) 𝑤 = (1 - 𝑛) 𝑔 (𝑥)

which is linear for $𝑤$ .

Practice problems

https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx
2016. Calculus, p. 665 (§9.5 exercises 23–25)

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