derivatives of solution of first order ODE

Suppose that f is a continuously differentiable function defined on an open subset E of 2, i.e. it has on E the continuousMathworldPlanetmath partial derivativesMathworldPlanetmathfx(x,y)  and  fy(x,y).

If y(x) is a solution of the ordinary differential equationMathworldPlanetmath

dydx=f(x,y), (1)

then we have

y(x)=f(x,y(x)), (2)
y′′(x)=fx(x,y(x))+fy(x,y(x))y(x) (3)

(see the chain ruleMathworldPlanetmath).  Thus there exists on E the second derivative y′′(x) which is also continuous.  More generally, we can infer the

Theorem.  If  f(x,y)  has in E the continuous partial derivatives up to the order n, then any solution y(x) of the differential equation (1) has on E the continuous derivatives y(i)(x) up to the order ( n+1.

Note 1.  The derivatives y(i)(x) are got from the equation (1) via succesive differentiations.  Two first ones are (2) and (3), and the next two ones, with a simpler notation:


Note 2.  It follows from (3) that the curve

fx(x,y)+fy(x,y)f(x,y)= 0 (4)

is the locus of the inflexion points of the integral curves of (1), or more exactly, the locus of the points where the integral curves have with their tangentsPlanetmathPlanetmath a contact of order ( more than one.  The curve (4) is also the locus of the points of tangency of the integral curves and their isoclines.


  • 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title derivatives of solution of first order ODE
Canonical name DerivativesOfSolutionOfFirstOrderODE
Date of creation 2013-03-22 18:59:14
Last modified on 2013-03-22 18:59:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 34A12
Classification msc 34-00
Related topic SolutionsOfOrdinaryDifferentialEquation
Related topic InflexionPoint