(Liouville’s formula)Derive (11) for the general case (i.e., where n need not equal 2), by...

(Liouville’s formula)

Derive (11) for the general case (i.e., where n need not equal 2), by showing that W'(x) is the sum of n determinants where the jth one is obtained from the W determinant by dif­ferentiating the jth row and leaving the other rows unchanged. Show that each of these n determinants, except the nth one, has two identical rows and hence vanishes, so that

In the last row, substitute u(n)(x)= −p1(x)u(n−1)(x) − • • • −pn(x)u(x) from (10), again omit vanishing determi­nants, and again obtain (5.1) and hence the solution (11). HINT: You may use the various properties of determinants, given in Section 10.4.