Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS

(iv) Iff(x) is the sum or product of any of the above then tryyp(x)asthe

sum or product of the corresponding individual trial functions.

It should be noted that this method fails if any term in the assumed trial

function is also contained within the complementary functionyc(x). In such a

case the trial function should be multiplied by the smallest integer power ofx

such that it will then contain no term that already appears in the complementary

function. The undetermined coefficients in the trial function can now be found

by substitution into (15.8).

Three further methods that are useful in finding the particular integralyp(x)are

those based on Green’s functions, the variation of parameters, and a change in the

dependent variable using knowledge of the complementary function. However,

since these methods are also applicable to equations with variable coefficients, a

discussion of them is postponed until section 15.2.

Find a particular integral of the equation

d^2 y

dx^2

− 2

dy

dx

+y=ex.

From the above discussion our first guess at a trial particular integral would beyp(x)=bex.
However, since the complementary function of this equation isyc(x)=(c 1 +c 2 x)ex(as
in the previous subsection), we see thatexis already contained in it, as indeed isxex.
Multiplying our first guess by the lowest integer power ofxsuch that the result does not
appear inyc(x), we therefore tryyp(x)=bx^2 ex. Substituting this into the ODE, we find
thatb=1/2, so the particular integral is given byyp(x)=x^2 ex/2.

Solution method.If the RHS of an ODE contains only functions mentioned at the

start of this subsection then the appropriate trial function should be substituted

into it, thereby fixing the undetermined parameters. If, however, the RHS of the

equation is not of this form then one of the more general methods outlined in sub-

sections 15.2.3–15.2.5 should be used; perhaps the most straightforward of these is

the variation-of-parameters method.

15.1.3 Constructing the general solutionyc(x)+yp(x)

As stated earlier, the full solution to the ODE (15.8) is found by adding together

the complementary function and any particular integral. In order to illustrate

further the material discussed in the last two subsections, let us find the general

solution to a new example, starting from the beginning.