second-order linear ODE with constant coefficients

Let’s consider the ordinary second-order linear differential equation

d2ydx2+adydx+by= 0 (1)

which is homogeneousPlanetmathPlanetmathPlanetmath ( and the coefficients a,b of which are constants.  As mentionned in the entry “finding another particular solution of linear ODE”, a simple substitution makes possible to eliminate from it the addend containing first derivativeMathworldPlanetmath of the unknown function.  Therefore we concentrate upon the case  a=0.  We have two cases depending on the sign of  b=±k2.

1.  b>0.  We will solve the equation

d2ydx2+k2y= 0. (2)

Multiplicating both addends by the expression 2dydx it becomes

2dydxd2ydx2+2k2ydydx= 0,

where the left hand side is the derivativePlanetmathPlanetmath of (dydx)2+k2y2.  The latter one thus has a constant value which must be nonnegative; denote it by k2C2.  We then have the equation

(dydx)2=k2(C2-y2). (3)

After taking the square root and separating the variables it reads


Integrating (see the table of integrals) this yields


where x0 is another constant.  Consequently, the general solution of the differential equation (2) may be written

y=Csink(x-x0) (4)

in which C and x0 are arbitrary real constants.

If one denotes  Ccoskx0=C1  and -Csinkx0=C2, then (4) reads

y=C1sinkx+C2coskx. (5)

Here, C1 and C2 are arbitrary constants.  Because both sinkx and coskx satisfy the given equation (2) and are linearly independentMathworldPlanetmath, its general solution can be written as (5).

2.  b<0.  An analogical treatment of the equation

d2ydx2-k2y= 0. (6)

yields for it the general solution

y=C1ekx+C2e-kx (7)

(note that one can eliminate the square root from the equation y±y2+C=Cekx and its “inverted equation” yy2+C=-CCe-kx).  The linear independence of the obvious solutions e±kx implies also the linear independence of coshkx and sinhkx and thus allows us to give the general solution also in the alternative form

y=C1sinhkx+C2coshkx. (8)

Remark.  The standard method for solving a homogeneous ( ordinary second-order linear differential equation (1) with constant coefficients is to use in it the substitution

y=erx (9)

where r is a constant; see the entry “second order linear differential equation with constant coefficients”.  This method is possible to use also for such equations of higher order.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.1.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title second-order linear ODE with constant coefficients
Canonical name SecondorderLinearODEWithConstantCoefficients
Date of creation 2014-03-01 17:02:54
Last modified on 2014-03-01 17:02:54
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 34A05