Leibniz’s Linear differential equation

A standard differential equation, introduced by a German mathematician Gottfried Wilhelm Leibniz (or Leibnitz) is called the Leibnitz’s (or Leibniz’s) linear differential equation.

Introduction

Leibniz (or Leibnitz) introduced a standard form linear differential equation of the first order and first degree.

$\dfrac{dy}{dx}+Py$ $\,=\,$ $Q$

It is defined in terms of two variables $x$ and $y$. In this equation, $P$ and $Q$ are the functions in terms of a variable $x$.

Gottfried Wilhelm Leibnitz derived a solution for this first order linear differential equation. Hence, this differential equation is called the Leibnitz’s linear differential equation or Leibniz’s linear differential equation. The following equation is the solution for the Leibnitz’s linear differential equation.

$ye^{\displaystyle \int{P}\,dx}$ $\,=\,$ $\displaystyle \int{Qe^{\displaystyle \int{P}}}\,dx$ $+$ $c$

On both sides of the solution, there is a factor in integration form and it is $e^{\displaystyle \int{P}\,dx}$. It is called the integrating factor, simply written as $I.F$.

$I.F \,=\, e^{\displaystyle \int{P}\,dx}$

Hence, the solution of the Leibniz’s linear differential equation is also written as follows.

$\implies$ $y(I.F)$ $\,=\,$ $\displaystyle \int{Q(I.F)}\,dx$ $+$ $c$

Solution

Learn how to prove the solution for the Leibnitz’s linear different equation.

Problems

List of the questions on the first order linear differential equations with solutions.