second order ordinary differential equation

A second order ordinary differential equationF(x,y,dydx,d2ydx2)=0  can often be written in the form

d2ydx2=f(x,y,dydx). (1)

By its general solution one means a functionxy=y(x)  which is at least on an interval twice differentiableMathworldPlanetmathPlanetmath and satisfies


By setting  dydx:=z,  one has  d2ydx2=dzdx,  and the equation (1) reads  dzdx=f(x,y,z).  It is easy to see that solving (1) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( with solving the system of simultaneous first order ( differential equationsMathworldPlanetmath

{dydx=z,dzdx=f(x,y,z), (2)

the so-called normal system of (1).

The system (2) is a special case of the general normal system of second order, which has the form

{dydx=φ(x,y,z),dzdx=ψ(x,y,z), (3)

where y and z are unknown functions of the variable x.  The existence theoremMathworldPlanetmath of the solution

{y=y(x),z=z(x) (4)

is as follows; cf. the Picard–Lindelöf theorem (

Theorem.  If the functions φ and ψ are continuousMathworldPlanetmathPlanetmath and have continuous partial derivativesMathworldPlanetmath with respect to y and z in a neighbourhood of a point  (x0,y0,z0),  then the normal system (3) has one and (as long as |x-x0| does not exceed a certain ) only one solution (4) which satisfies the initial conditionsMathworldPlanetmathy(x0)=y0,z(x0)=z0.  The functions (4) are continuously differentiable in a neighbourhood of x0.


  • 1 E. Lindelöf: Differentiali- ja integralilasku III 1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title second order ordinary differential equation
Canonical name SecondOrderOrdinaryDifferentialEquation
Date of creation 2013-03-22 18:35:39
Last modified on 2013-03-22 18:35:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Topic
Classification msc 34A05
Defines normal system
Defines normal system of second order