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 straight forwardly 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 never equal to zero we simply 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.
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
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
Integrating factor: .
Multiplying the differential equation with 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: