second-order linear ODE with constant coefficients
and the coefficients 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
derivative of the unknown function. Therefore we
concentrate upon the case . We have two cases
depending on the sign of .
. . We will solve the equation
Multiplicating both addends by the expression it becomes
where the left hand side is the derivative of . The latter one thus has a constant value which must be nonnegative; denote it by . We then have the equation
After taking the square root and separating the variables it reads
Integrating (see the table of integrals) this yields
in which and are arbitrary real constants.
If one denotes and , then (4) reads
. . An analogical treatment of the equation
yields for it the general solution
(note that one can eliminate the square root from the equation and its “inverted equation” ). The linear independence of the obvious solutions implies also the linear independence of and and thus allows us to give the general solution also in the alternative form
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
where 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|
|Date of creation||2014-03-01 17:02:54|
|Last modified on||2014-03-01 17:02:54|
|Last modified by||pahio (2872)|