Technique for solving linear ordinary differential equations
Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution is known and a second linearly independent solution is desired. The method also applies to n-th order equations. In this case the ansatz will yield an (n−1)-th order equation for .
Second-order linear ordinary differential equations[edit]
An example[edit]
Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE)
where
are real non-zero coefficients. Two linearly independent solutions for this ODE can be straightforwardly found using
characteristic equations except for the case when the
discriminant,
, vanishes. In this case,
from which only one solution,
can be found using its characteristic equation.
The method of reduction of order is used to obtain a second linearly independent solution to this differential equation using our one known solution. To find a second solution we take as a guess
where
is an unknown function to be determined. Since
must satisfy the original ODE, we substitute it back in to get
Rearranging this equation in terms of the derivatives of
we get
Since we know that is a solution to the original problem, the coefficient of the last term is equal to zero. Furthermore, substituting into the second term's coefficient yields (for that coefficient)
Therefore, we are left with
Since is assumed non-zero and is an exponential function (and thus always non-zero), we have
This can be integrated twice to yield
where
are constants of integration. We now can write our second solution as
Since the second term in is a scalar multiple of the first solution (and thus linearly dependent) we can drop that term, yielding a final solution of
Finally, we can prove that the second solution found via this method is linearly independent of the first solution by calculating the Wronskian
Thus is the second linearly independent solution we were looking for.
General method[edit]
Given the general non-homogeneous linear differential equation
and a single solution
of the homogeneous equation [
], let us try a solution of the full non-homogeneous equation in the form:
where
is an arbitrary function. Thus
and
If these are substituted for , , and in the differential equation, then
Since is a solution of the original homogeneous differential equation, , so we can reduce to
which is a first-order differential equation for
(reduction of order). Divide by
, obtaining
The integrating factor is .
Multiplying the differential equation by the integrating factor , the equation for can be reduced to
After integrating the last equation, is found, containing one constant of integration. Then, integrate to find the full solution of the original non-homogeneous second-order equation, exhibiting two constants of integration as it should:
See also[edit]
References[edit]