Variation of Parameters. Consider the following method of

Chapter 2, Problem 38

(choose chapter or problem)

Variation of Parameters. Consider the following method of solving the general linearequation of first order:y+ p(t)y = g(t). (i)(a) If g(t) = 0 for all t, show that the solution isy = A exp p(t) dt, (ii)where A is a constant.(b) If g(t) is not everywhere zero, assume that the solution of Eq. (i) is of the formy = A(t) exp p(t) dt, where A is now a function of t. By substituting for y in the given differential equation,show that A(t) must satisfy the conditionA(t) = g(t) exp p(t) dt. (iv)(c) Find A(t) from Eq. (iv). Then substitute for A(t) in Eq. (iii) and determine y. Verifythat the solution obtained in this manner agrees with that of Eq. (33) in the text. Thistechnique is known as the method of variation of parameters; it is discussed in detail inSection 3.6 in connection with second order linear equations.