The Method of Judicious Guessing

A serious disadvantage of the method of variation of parameters is that the integrations required are often quite difficult. In certain cases, it is usually much simpler to guess a particular solution. In this section we will establish a systematic method for guessing solutions of the equation

                                                                    (1)

where , and are constants, and has one of several special forms.

PART I: Consider first the differential equation

                                               (2)

We seek a function such that the three functions , and add up to a given polynomial of degree . One can show that the differential equation (2) has a solution of the form

                                         (3)

EXAMPLE: Find a particular solution of the equation

                                                                  (4)


PART II: Consider now the differential equation

                                          (5)

One can show that (5) has a particular solution of the form



Equivalently,



EXAMPLE: Find the general solution of the equation

(a)                       (b)


EXAMPLE: Find the general solution of the equation



EXAMPLE: Find a particular solution of the equation



PART III: Let be a particular solution of the equation

                                                (6)

Since



the real part of the right-hand side of (6) is



while the imaginary part is



One can show that



is a solution of



while



is a solution of



EXAMPLE: Find a particular solution of the equation

                                                                   (7)


EXAMPLE: Find a particular solution of the equation

                                                                 (11)


EXAMPLE: Find a particular solution of the equation

                                                          (12)