on inhomogeneous second-order linear ODE with constant coefficients
Let’s consider solving the ordinary second-order linear differential equation
which is inhomogeneous (http://planetmath.org/HomogeneousLinearDifferentialEquation), i.e. .
particular solution (http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation)
of the inhomogeneous equation (1). A latter one can
always be gotten by means of the variation of parameters, but
in many cases there exist simpler ways to find a particular
solution of (1).
: is a nonzero constant function
. In this case, apparently is
a solution of (1), supposing that . If
but , a particular solution is .
If , a solution is gotten via two consecutive
: is a polynomial function of degree
. Now (1) has as solution a polynomial which can be
found by using indetermined coefficients. If ,
the polynomial is of degree and is uniquely determined.
If and , the degree of the polynomial
is and its constant term is arbitrary. If
the polynomial is of degree and is
gotten via two integrations.
: Let in (1) be of the form with , , constants. We try to find a solution of the same form and put into (1) the expression
Then the left hand side of (1) attains the form
This must equal , i.e. we have the conditions
These determine uniquely the values of and provided that the determinant
Unless we have , the equation (4) has no solution of the form (3), since
The right hand side coincides with the right hand side of (4) iff and , and thus (4) has the solution
: Let in (1) now be where and are constants. Denote the left hand side of (1) briefly . We seek again a solution of the same form as .
First we have
Thus can be determined from the condition . If , i.e. is not a root of the characteristic equation corresponding the homogeneous equation (2), then we obtain the particular solution
of the inhomogeneous equation (1).
If , then and satisfy the homogeneous equation . Now we may start from the identity
and differentiate it with respect to . Changing again the order of differentiations we can write first
and differentiating anew,
If is a simple root of the equation , i.e. if but , then makes the right hand side of (6) to , which equals to by choosing . Then we have found the particular solution
We have still to handle the case when is the double root of the equation and thus . Putting into (7), the right hand side reduces to ; this equals to when choosing . So we have the particular solution
of the given inhomogeneous equation.
: Suppose that in (1) the right hand side is a sum of several functions,
and one can find a particular solution for each of the equations
Then evidently the sum is a
particular solution of the equation (8).
- 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||on inhomogeneous second-order linear ODE with constant coefficients|
|Date of creation||2014-03-05 16:25:57|
|Last modified on||2014-03-05 16:25:57|
|Last modified by||pahio (2872)|