The Bernoulli differential equation is a type of nonlinear first-order ordinary differential equation (ODE) that has the form: (dy/dx) + P(x)y = Q(x)y^n

Here:

y is the dependent variable,

x is the independent variable,

P(x) and Q(x) are functions of x,

n is a constant.

The key characteristic that distinguishes the Bernoulli differential equation is the presence of the term Q(x)y^n, which makes it nonlinear. The equation can be transformed into a linear first-order ODE by using a substitution.

A common transformation involves dividing both sides of the equation by y^n to obtain: (1/y^n)(dy/dx) + P(x)y^(1-n) = Q(x)

Now, let v = y^(1-n), and differentiate with respect to x: dv/dx = (1-n)y^(-n)(dy/dx)

Substitute this into the transformed equation: (1/(1-n))(dv/dx) + P(x)v = Q(x)

This transformed equation is now a linear first-order ODE, and standard methods for solving linear ODEs can be applied. Once the solution for v is found, it can be substituted back to find the solution for y.

It's worth noting that the Bernoulli differential equation is named after the Swiss mathematician Jacob Bernoulli, who worked on various mathematical topics during the 17th and 18th centuries. The equation is encountered in diverse fields, including physics and biology, and its solution often involves techniques from linear differential equations.