method of integrating factors

The method of integrating factors is in principle a means for solving ordinary differential equationsMathworldPlanetmath of first .  It has not great practical significance, but is theoretically important.

Let us consider a differential equation solved for the derivativePlanetmathPlanetmath y of the unknown function and write the equation in the form

X(x,y)dx+Y(x,y)dy= 0. (1)

We assume that the functions X and Y have continuousMathworldPlanetmathPlanetmath partial derivativesMathworldPlanetmath in a region R of 2.

If there is a solution of (1) which may be expressed in the form


with f having continuous partial derivatives in R and with C an arbitrary constant, then it’s not difficult to see that such an f satisfies the linear partial differential equation

Xfy-Yfx= 0. (2)

Conversely, every non-constant solution f of (2) gives also a solution  f(x,y)=C  of (1).  Thus, solving (1) and solving (2) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( tasks.

It’s straightforward to show that if  f0(x,y)  is a non-constant solution of the equation (2), then all solutions of this equation are  F(f0(x,y))  where F is a freely chosen function with (mostly) continuous derivative.

The connection of the equations (1) and (2) may be presented also in another form.  Suppose that  f(x,y)=C  is any solution of (1).  Then (2) implies the proportion equation


If we denote the common value of these two ratios by  μ(x,y)=μ,  then we have


This gives to the differentialMathworldPlanetmath of the function f the expression


We see that  μ(x,y)  is the integrating factor or Euler multiplicator of the given differential equation (1), i.e. the left hand side of (1) turns, when multiplied by  μ(x,y),  to an exact differential (

Conversely, any integrating factor μ of (1), i.e. such that  μXdx+μYdy  is the differential of some function f, is easily seen to determine the solutions of the form  f(x,y)=C  of (1).  Altogether, solving the differential equation (1) is equivalent with finding an integrating factor of the equation.

When an integrating factor μ of (1) is available, the solution function f can be gotten from the line integral


along any curve γ connecting an arbitrarily chosen point  P0=(x0,y0)  and the point  P=(x,y)  in the region R.

Note.  In general, it’s very hard to find a suitable integrating factor.  One special case where such can be found, is that X and Y are homogeneous functions of same degree ( then the expression 1xX+yY is an integrating factor.

Example.  In the differential equation

(x4+y4)dx-xy3dy= 0

we see that  X=:x4+y4  and  Y=:-xy3  both define a homogeneous function of degree ( 4.  Thus we have the integrating factor  μ=:1x5+xy4-xy4=1x5,  and the left hand side of the equation

(1x+y4x5)dx-y3x4dy= 0

is an exact differential.  We can integrate it along the broken line, first from  (1, 0)  to  (x, 0)  and then still to  (x,y),  obtaining


So the general solution of the given differential equation is



  • 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title method of integrating factors
Canonical name MethodOfIntegratingFactors
Date of creation 2013-03-22 16:31:48
Last modified on 2013-03-22 16:31:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 21
Author pahio (2872)
Entry type Topic
Classification msc 35-00
Classification msc 34-00
Related topic ErnstLindelof
Defines integrating factor
Defines Euler multiplicator