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Reduction of order

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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

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An example

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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

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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

One integrating factor is given by , and because

this integrating factor can be more neatly expressed as 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

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References

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  • Boyce, William E.; DiPrima, Richard C. (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). Hoboken, NJ: John Wiley & Sons, Inc. ISBN 978-0-471-43338-5.
  • Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
  • Eric W. Weisstein, Second-Order Ordinary Differential Equation Second Solution, From MathWorld—A Wolfram Web Resource.