Reduction of order

Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations.

is known and a second linearly independent solution

The method also applies to n-th order equations.

In this case the ansatz will yield an (n−1)-th order equation for

Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation.

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,

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

is an unknown function to be determined.

must satisfy the original ODE, we substitute it back in to get

Rearranging this equation in terms of the derivatives of

is a solution to the original problem, the coefficient of the last term is equal to zero.

is assumed non-zero and

is an exponential function (and thus always non-zero), we have

This can be integrated twice to yield

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

is the second linearly independent solution we were looking for.

Given the general non-homogeneous linear differential equation

], let us try a solution of the full non-homogeneous equation in the form:

is an arbitrary function.

is a solution of the original homogeneous differential equation,

which is a first-order differential equation for

, and because this integrating factor can be more neatly expressed as

Multiplying the differential equation by the integrating factor

is found, containing one constant of integration.

to find the full solution of the original non-homogeneous second-order equation, exhibiting two constants of integration as it should: