Bernoulli differential equation

Equation

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

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 $y^{\displaystyle n}$.

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 $y^{\displaystyle n}$ also forms another mathematical expression as follows.

$Qy^{\displaystyle n}$

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

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

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

$y\,’+Py$ $\,=\,$ $Qy^{\displaystyle n}$

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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