# Bernoulli differential equation

## Equation

$\dfrac{dy}{dx}+P(x)y$ $\,=\,$ $Q(x)y^n$

A differential equation in the above form is called the Bernoulli differential equation.

### Introduction

Let $x$ and $y$ be two variables, $n$ be a constant and, $P(x)$ and $Q(x)$ be two mathematical functions in $x$ algebraically. The functions $P(x)$ and $Q(x)$ are simply denoted by $P$ and $Q$ for our convenience. The variable $y$ raised to the power $n$ forms a power function $y^n$.

The sum of the derivative of $y$ with respect to $x$ and the product of the function $P$ and variable $y$ forms a mathematical expression in the following form.

$\dfrac{dy}{dx}+Py$

Similarly, the product of the function $Q$ and power function $y^n$ also forms another mathematical expression as follows.

$Qy^n$

Suppose, the two mathematical expressions are equal, the mathematical equation is called the Bernoulli’s differential equation.

$\dfrac{dy}{dx}+Py$ $\,=\,$ $Qy^n$

It is also written in the following mathematical form in calculus.

$y\,’+Py$ $\,=\,$ $Qy^n$

Jacob Bernoulli introduced this special nonlinear differential equation but Gottfried Wilhelm Leibniz solved it by reducing it to linear form differential equation.

#### Solution

Learn how to solve the Bernoulli’s differential equation by reducing it to Leibnitz’s linear differential equation.

##### Problems

List of the Bernoulli’s differential equations questions with step-by-step solutions to learn how to solve them mathematically.

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Jun 26, 2023
Jun 23, 2023

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