ODE types reductible to the variables separable case

There are certain of non-linear ordinary differential equationsMathworldPlanetmath of first order (http://planetmath.org/ODE) which may by a suitable substitution be to a form where one can separate (http://planetmath.org/SeparationOfVariables) the variables.

This means the equation of the form


where X and Y are two homogeneous functions of the same degree (http://planetmath.org/HomogeneousFunction).  Therefore, if the equation is written as


its right hand side is a homogeneous function of degree 0, i.e. it depends only on the ratio y:x, and has thus the form

dydx=f(yx). (1)

Accordingly, if this ratio is constant, then also dydx is constant; thus all lines   yx= constant  are isoclines of the family of the integral curves which intersect any such line isogonally.

We can infer as well, that if one integral curve is represented by  x=x(t),  y=y(t),  then also  x=Cx(t),  y=Cy(t)  an integral curve for any constant C.  Hence the integral curves are homotheticMathworldPlanetmath with respect to the origin; therefore some people call the equation (1) a similarity equation.

For generally solving the equation (1), make the substitution


The equation takes the form

t+xdtdx=f(t) (2)

which shows that any root (http://planetmath.org/Equation) tν of the equality  f(t)=t  gives a singular solutiony=tνx. The variables in (2) may be :


Thus one obtains  ln|x|=dtf(t)-t+lnC, whence the general solution of the homogeneous differential equation (1) is in a parametric form


II.  Equation of the form  y= f(ax+by+c)

It’s a question of the equation

dydx=f(ax+by+c), (3)

where a, b and c are given constants.  If  ax+by is constant, then dydx is constant, and we see that the lines  ax+by= constant  are isoclines of the intgral curves of (3).


ax+by+c:=u (4)

be a new variable.  It changes the equation (3) to

dudx=a+bf(u). (5)

Here, one can see that the real zeros u of the right hand side yield lines (4) which are integral curves of (3), and thus we have singular solutions.  Moreover, one can separate the variables in (5) and integrate, obtaining x as a function of u.  Using still (4) gives also y.  The general solution is


Example.  In the nonlinear equation


which is of the type II, one cannot separate the variables x and y.  The substitution  x-y:=u  converts it to


where one can separate the variables.  Since the right hand side has the zeros  u=±1,  the given equation has the singular solutions y given by  x-y=±1.  Separating the variables x and u, one obtains




Accordingly, the given differential equation has the parametric solution



  • 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title ODE types reductible to the variables separable case
Canonical name ODETypesReductibleToTheVariablesSeparableCase
Date of creation 2013-03-22 18:06:36
Last modified on 2013-03-22 18:06:36
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Topic
Classification msc 34A09
Classification msc 34A05
Related topic SeparationOfVariables
Related topic ODETypesSolvableByTwoQuadratures
Related topic TheoryForSeparationOfVariables
Defines homogeneous differential equation
Defines similarity equation