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.
Section 1.2 Gauss-Jordan Elimination There are three legal elementary row operations: (1) Multiply a row by a nonzero constant. Example: 2R 1 would multiply every entry in the first row of a matrix by 2. (2) Switch two rows. R ↔ R Example: 1 2 would interchange the elements in the first and second rows. (3) Add a multiple of one row to another. R + 2R Example: 1 2 would add twice the elements in the second row to the elements in the first row. Note that whatever is written first is what changes. Gauss-Jordan Elimination is made up of any combination of these row 10 a operations. Our goal is to get 1 b 0 x 010 y or 1 z Example: Use Gauss-Jordan
