PROJECT. Reduction of Order. This is of practical interest

Chapter 3, Problem 3.2

(choose chapter or problem)

PROJECT. Reduction of Order. This is of practical interest since a single solution of an ODE can often be guessed. For second order. see Example 7 in Sec. 2.1. (a) How could you reduce the order of a linear constant-coefficient ODE if a solution is known? (b) Extend the method to a variable-coefficient ODE .1'''' + P2(xly" + PI(X)Y' + Po(x)y = o. Assuming a solution YI to be known, show that another solution is Y2(X) = U(X)YI(X) with u(x) = J z(x) dx and .:: obtained by solving )"1'::" + (3y; + P2YI)'::' + (3y~ + 2P2Y ; + PIYI)':: = O. (e) Reduce x3)"'" - 3x\" + (6 - X2)X/ - (6 - X2)" = O. using Yl = x (perhaps obtainable by inspection).