### Integrating Factors and Repeated Roots of the Characteristic Equation

Howard Dwyer, William Green
2012 CODEE Journal
Most texts on elementary differential equations solve homogeneous constant coefficient linear equations by introducing the characteristic equation; once the roots of the characteristic equation are known the solutions to the differential equation follow immediately, unless there is a repeated root. In this paper we show how an integrating factor can be used to find all of the solutions in the case of a repeated root without depending on an assumption about the form that these solutions will
more » ... solutions will take. We also show how an integrating factor can be used to explain the "extra" power of t which appears in the trial form of the solution when using the method of undetermined coefficients on a nonhomogeneous equation in the case where the right hand side is a polynomial multiple of the corresponding homogeneous solution. Motivation and Intuition Constant coefficient, linear differential equations are well-studied in introductory differential equation classes. The standard method is to use an ansatz to transform the differential equation into a polynomial algebraic equation, which is easily solved. The standard approach works well when the algebraic equation doesn't have repeated roots. We offer an alternative approach to explain the form of solutions obtained from repeated roots. We further show that our approach, which is based on simple first order methods, applies equally well to homogeneous and nonhomogeneous equations with repeated roots. Consider the first order, constant coefficient linear differential equation, y ′ (t) + r 1 y(t) = g(t). (1.1) Here r 1 is a constant and g(t) is an arbitrary function. Every equation of this form is solved by use of the integrating factor µ(t) = e r 1 t . Multiplying both sides of the equation by µ(t), CODEE Journal • Article Digital Library