Bernoulli Differential Equation

This lecture covers the Bernoulli differential equation, which is a first-order differential equation of the form y'(x) = g(x) y(x) + f(x) (y(x))^(n+0 and n+1). The historical background of this equation is discussed, including Jakob Bernoulli's efforts to solve it. Two elegant solutions proposed by his brother Johann are presented. The lecture also explores the solution methods, such as using variable changes and seeking solutions in the form y(x) = u(x) v(x). Additionally, linear homogeneous differential equations of order n are examined, along with the theorem stating properties of solutions. Examples and the general solution for specific cases are provided.