equation y′′=f(x)

A simple special case of the second order linear differential equation with constant coefficients is

d2ydx2=f(x) (1)

where f is continuousMathworldPlanetmath.  We obtain immediately  dydx=C1+f(x)𝑑x,

y=C1x+C2+(f(x)𝑑x)𝑑x. (2)

A particular solution y(x) of (1) satisfying the initial conditionsMathworldPlanetmath


is obtained more simply by integrating (1) twice between the limits (http://planetmath.org/DefiniteIntegral) x0 and x, thus getting


But here, the two first addends are the first terms of the Taylor polynomialMathworldPlanetmath of y(x), expanded by the powers of x-x0, whence the double integral is the corresponding remainder term (http://planetmath.org/RemainderVariousFormulas)


Hence the particular solution can be written with the simple integral as

y(x)=y0+y0(x-x0)+x0xf(t)(x-t)𝑑t. (3)

The result may be generalised for the nth order (http://planetmath.org/ODE) differential equationMathworldPlanetmath

dnydxn=f(x) (4)

with corresponding n initial conditions:

y(x)=y0+y0(x-x0)+y0′′2!(x-x0)2++y0(n-1)(n-1)!(x-x0)n-1+1(n-1)!x0xf(t)(x-t)n-1𝑑t. (5)
Title equation y′′=f(x)
Canonical name EquationYFx
Date of creation 2013-03-22 18:35:33
Last modified on 2013-03-22 18:35:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Topic
Classification msc 34A30
Classification msc 34-01