The document focuses on Bernoulli equations, which are first-order differential equations characterized by the form y′(x) + p(x)y = r(x)y^α. It explains that these equations are separable when α = 0 and linear when α = 1. Additionally, it includes examples demonstrating how to solve specific Bernoulli equations.

1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #8
BERNOULLI EQUATION:
A Bernoulli equation is a first order equation
𝑦′(𝑥) + 𝑃(𝑥)𝑦 = 𝑅(𝑥)𝑦 𝛼
In which 𝛼 is a real number.
A Bernoulli equation is separable if 𝛼 = 0 and linear if 𝛼 = 1. About 1696, Leibniz
showed that a Bernoulli equation with 𝛼 ≠ 1 transforms to a linear equation under
the change of variables
𝑣 = 𝑦1−𝛼
EXAMPLE#1: Solve
𝑑𝑦
𝑑𝑥
+
1
𝑥
𝑦 = 3𝑥2
𝑦3
EXAMPLE#2: Solve
𝑑𝑦
𝑑𝑥
+
1
𝑥
𝑦 =
2
𝑥3
𝑦−4/3