Since variation of parameters is generally complicated, it seems worthwhile

Since variation of parameters is generally complicated, it seems worthwhile to try to extend the other method. Find out experimentally for what ODEs this is possible and for what not. Work backward, solving ODEs with a CAS and then looking whether the solution could be obtained by undetermined coefficients. For example, consider

y"' - 3y" + 3y' - y = x1/2ex

and

x3y"' + x2y" - 2xy' + 2y = x3 ln x.