Reduction of order

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Reduction of order 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 a (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 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.

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

.

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:

.

See also[edit]

References[edit]