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 (http://planetmath.org/HomogeneousLinearDifferentialEquation) 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

dy±C2-y2=kdx.

Integrating (see the table of integrals) this yields

arcsinyC=k(x-x0)

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 (http://planetmath.org/HomogeneousLinearDifferentialEquation) 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.

References

  • 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