Inexact differential equation
# Inexact differential equation

Differential equations :

Scope :

Fields

Natural sciencesEngineering

Astronomy

Physics

Chemistry

Biology

Geology

Applied mathematics

Continuum mechanics

Chaos theory

Dynamical systems

Social sciences

Economics

Population dynamics

List of named differential equations :

Classification :

Types

Ordinary

Partial

Differential-algebraic

Integro-differential

Fractional

Linear

Non-linear

By variable type

Dependent and independent variables

Autonomous

Coupled / Decoupled

Exact

Homogeneous / Nonhomogeneous

Features

Order

Operator

Notation :

Relation to processes

Difference (discrete analogue)

Stochastic

Stochastic partial

Delay :

Solution :

Existence and uniqueness

Picard–Lindelöf theorem

Peano existence theorem

Carathéodory's existence theorem

Cauchy–Kowalevski theorem :

General topics

Initial conditions

Boundary values

Dirichlet

Neumann

Robin

Cauchy problem

Wronskian

Phase portrait

Lyapunov / Asymptotic / Exponential stability

Rate of convergence

Series / Integral solutions

Numerical integration

Dirac delta function :

Solution methods

Inspection

Method of characteristics

Euler

Exponential response formula

Finite difference (Crank–Nicolson)

Finite element

Infinite element

Finite volume

Galerkin

Petrov–Galerkin

Green's function

Integrating factor

Integral transforms

Perturbation theory

Runge–Kutta

Separation of variables

Undetermined coefficients

Variation of parameters :

People :

List

Isaac Newton

Gottfried Leibniz

Jacob Bernoulli

Leonhard Euler

Józef Maria Hoene-Wroński

Joseph Fourier

Augustin-Louis Cauchy

George Green

Carl David Tolmé Runge

Martin Kutta

Rudolf Lipschitz

Ernst Lindelöf

Émile Picard

Phyllis Nicolson

John Crank :

vte :

An inexact differential equation is a differential equation of the form (see also: inexact differential)

    $M(x,y)\,dx+N(x,y)\,dy=0,{\text{ where }}{\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}.$

The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739.

## Solution method

In order to solve the equation, we need to transform it into an exact differential equation. In order to do that, we need to find an integrating factor $\mu$ to multiply the equation by. We'll start with the equation itself. $M\,dx+N\,dy=0$, so we get $\mu M\,dx+\mu N\,dy=0$. We will require $\mu$ to satisfy ∂μM/∂y=∂μN/∂x {\textstyle {\frac {\partial \mu M}{\partial y}}={\frac {\partial \mu N}{\partial x}}}. We get

    ${\frac {\partial \mu }{\partial y}}M+{\frac {\partial M}{\partial y}}\mu ={\frac {\partial \mu }{\partial x}}N+{\frac {\partial N}{\partial x}}\mu .$

After simplifying we get

    $M\mu _{y}-N\mu _{x}+(M_{y}-N_{x})\mu =0.$

Since this is a partial differential equation, it is mostly extremely hard to solve, however in some cases we will get either $\mu (x,y)=\mu (x)$ or $\mu (x,y)=\mu (y)$, in which case we only need to find $\mu$ with a first-order linear differential equation or a separable differential equation, and as such either

    $\mu (y)=e^{-\int {{\frac {M_{y}-N_{x}}{M}}\,dy}}$

or

    $\mu (x)=e^{\int {{\frac {M_{y}-N_{x}}{N}}\,dx}}.$
