theory for separation of variables

The first order ( ordinary differential equationMathworldPlanetmath where one can separate the variables has the form where dydx may be expressed as a productPlanetmathPlanetmath or a quotient of two functions (, one of which depends only on x and the other on y.  Such an equation may be written e.g. as

dydx=Y(y)X(x)ordxdy=X(x)Y(y). (1)

We notice first that if Y(y) has real zeroes ( y1,y2,, then the equation (1) has the constant solutions  y:=y1,y:=y2,  and thus the lines  y=y1,y=y2,  are integral curves.  Similarly, if X(x) has real zeroes  x1,x2,, one has to include the lines  y=y1,y=y2,  to the integral curves.  All those lines the xy-plane into the rectangular regions.  One can obtain other integral curves only inside such regions where the derivativePlanetmathPlanetmath dydx attains real values.

Let R be such a region, defined by


and let us assume that the X(x) and Y(y) are real, continuousMathworldPlanetmathPlanetmath and distinct from zero in R.  We will show that any integral curve of the differential equation (1) is accessiblePlanetmathPlanetmath by two quadratures.

Let γ be an integral curve passing through the point  (x0,y0)  of the region R.  By the above assumptionsPlanetmathPlanetmath, the derivative dydx maintains its sign on the curve γ so long γ is inside R, which is true on a neighbourhood N of  x0, contained in  [a,b].  This implies that as x runs the intervalN,  it defines the ordinate y of γ uniquely as a monotonic functionyy(x)  which satisfies the equation (1):


The last equation may be written

y(x)Y(y(x))=1X(x). (2)

Since X and Y don’t vanish in R, the denominators Y(y(x)) and X(x) are distinct from 0 on the interval N.  Therefore one can integrate both sides of (2) from x0 to an arbitrary value x on N, getting

x0xy(x)dxY(y(x))=x0xdxX(x). (3)

Because  y=y(x)  is continuous and monotonic on the interval N, it can be taken as new variable of integration ( in the left hand side of (3):  substitute  y(x):=y,  y(x)dx:=dy  and change the to  y(x0)=y0  and  y(x)=y.

  • Accordingly, the equality

    y0ydyY(y)=x0xdxX(x) (4)

    is valid, meaning that if an integral curve of (1) passes through the point  (x0,y0), the integral curve is represented by the equation (4) as long as the curve is inside the region R.

  • Additionally, it is possible to justificate that if  (x0,y0)  is an interior point of a region R where X(x) and Y(y) are real, continuous and 0, then one and only one integral curve of (1) passes through this point, the curve is regular (, and both x and y are monotonic on it.  N.B., the Lipschitz conditionMathworldPlanetmath for the right hand side of (1) is not necessary for the justification.

  • When the point  (x0,y0)  changes in the region R, (4) gives a family of integral curves which cover the region once.  The equations of these curves may be unified to the form

    dyY(y)=dxX(x), (5)

    which thus the general solution of the differential equation (1) in R.  Hence one can speak of the separation of variablesMathworldPlanetmath,

    dyY(y)=dxX(x), (6)

    and integration of both sides.


  • 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title theory for separation of variables
Canonical name TheoryForSeparationOfVariables
Date of creation 2013-03-22 18:37:43
Last modified on 2013-03-22 18:37:43
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Topic
Classification msc 34A09
Classification msc 34A05
Related topic InverseFunctionTheorem
Related topic ODETypesReductibleToTheVariablesSeparableCase