Advanced Calculus fi..

642 Advanced Calculus, Fifth EditionWhen repeated roots occur, the procedure has to be modified. If A is an eigenvalueof multiplicity k, one replaces the expression eh'u by eAtq(t) where q(t) is acolumn vector function whose components are polynomials of degree at most k - 1.It can be shown that n linearly independent solutions can be found in this form(with complex roots treated as previously). For details, see Section 6.22 of ODE.The method of Laplace transforms also handles all cases without difficulty (seeProblem 3 which follows).Nonhomogeneous equations. As in Section 9.5, we obtain a particular solutionx*(t) by the variation-of-parameters formula (9.37). The general solution is thenobtained as in Section 9.5 as2 , :9 = x - 2y + cost, 2 = -2x + y - sin t. The related homoge-EXAMPLE 3neous system is that of Example 1, and therefore we can takeWe also find'X-'(t)p(t) dt to be2 / ['-"et :;'I [:,I dt = - [1 1 e-3' (cos t +1sin t) dt2 1 er(cos t - sin t) drand therefore findThe general solution is- 7Ix = cle3' + c2e-' + -(3 cos t - sin t),103t 1y = -cle + c2e7' + -(7 cos t + sin t).10