Differential Equations and Linear Algebra, 4th Edition

10.3 Translation and Partial Fractions

As illustrated by Examples 1 and 2 of Section 10.2, the solution of a linear differential equation with constant coefficients can often be reduced to the matter of finding the inverse Laplace transform of a rational function of the form

$$R(s)={\displaystyle \frac{P(s)}{Q(s)}}$$ (1)

where the degree of P(s) is less than that of Q(s). The technique for finding ${\mathcal{L}}^{-1}\{R(s)\}$ is based on the same method of partial fractions that we use in elementary calculus to integrate rational functions. The following two rules describe the partial fraction decomposition of R(s), in terms of the factorization of the denominator Q(s) into linear factors and irreducible quadratic factors corresponding to the real and complex zeros, respectively, of Q(s