## Abstract

This paper shows a simple tabular procedure \added{derived from the method of undetermined coefficients} for finding a particular solution to differential equations of the form \sum_{j=0}^m a_j\frac{d^j y}{dx^j} = P(x)e^{\alpha{}x}. This procedure reduces the derivatives of the product of an arbitrary polynomial and an exponential to rows of constants representing the coefficients of the terms. The rows are each multiplied by aj and summed to produce a mth order differential equation such that its solution is the polynomial part of the particular solution of the above equation. Solving this corresponding differential equation determines the coefficients of the polynomial. The underlying algebra of this conversion and its formulaic implication are then discussed. Using the formula derived, the particular solution is found. This procedure is based on but different than the method of undetermined coefficients because while the method of undetermined coefficients requires substitution of a product of a polynomial, Q, and an exponential into the differential equation immediately, this procedure is derived from the examination of the substitution of the product of any function and an exponential. This allows for a richer understanding of the relationship between the differential equation for y and the differential equation for Q. Ultimately this method is better than the method of undetermined coefficients because it is more straightforward. In any case, both methods solve the same problem but KMO is faster.

## Faculty Sponsor

Charles E Bell

## Recommended Citation

Krohn, David; Mariño-Johnson, Daniel; and Ouyang, John Paul
(2014)
"The KMO Method for Solving Non-homogenous, mth Order Differential Equations,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 15:
Iss.
1, Article 8.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol15/iss1/8