finding another particular solution of linear ODE

Consider the homogeneousPlanetmathPlanetmathPlanetmathPlanetmath ( second-order linear ordinary differential equationMathworldPlanetmath

y′′+P(x)y+Q(x)y= 0. (1)

If one knows one particular solution (  y=y1(x)0  of (1), it’s possible to derive from it via two quadratures another solution  y2(x),  linearly independentMathworldPlanetmath on  y1(x);  thus one can write the general solution


of that homogeneous differential equation.

We will now show the derivation procedure.

We put

y=uv (2)

which renders (1) to

(v′′+Pv+Qv)u+(2v+Pv)u+u′′v= 0. (3)

Here one can choose  v:=y1(x), whence the first addend vanishes, and (3) gets the form

(2y1+Py1)u+y1u′′= 0. (4)

This equation may be written as  u′′u=-2y1y1-P, which is integrated to




A new integration results from this the general solution of (4):


Thus by (2), we have obtained the wanted other solution


which is clearly linearly independent on y_1(x).

Consequently, we can express the general solution of the differential equation (1) as


where C1 and C2 are arbitrary constants.

Remark.  The substitution


converts the equation (1) into the form

d2udx2+(Q-P24-P2)u= 0

not containing the derivativePlanetmathPlanetmath dudx.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title finding another particular solution of linear ODE
Canonical name FindingAnotherParticularSolutionOfLinearODE
Date of creation 2014-02-28 14:31:42
Last modified on 2014-02-28 14:31:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type AlgorithmMathworldPlanetmath
Classification msc 34A05