201. Solution of homogeneous linear equations. The equations considered in the previous section become homogeneous (p. 115) if f1 = ƒ2 = ƒs = ƒ1 = 0. We have then

= 0.

These equations have evidently the solution x = y = z =w= This we call the zero solution. We seek the condition that the coefficients must fulfill in order that other solutions also may exist. If we carry out the method of the previous section, we observe that the determinant equals zero in the numerator of every fraction which affords the value of one of the variables (§ 191). Thus if D is not equal to zero, the only solution of the above equations is the zero solution. This gives us the following

PRINCIPLE. A system of n linear homogeneous equations in n variables has a solution distinct from the zero solution only when the determinant of the system vanishes.

Whether a solution distinct from the zero solution always exists when the determinant of the system equals zero we shall not determine, as a complete discussion of the question would be beyond the scope of this chapter.

THEOREM. If x1, Y, Z1, w1 is a solution of equations (I) and k is any number, then kx,, ky,, kz, kw, is also a solution.

The proof of this theorem is evident on substituting kx1, etc., in equations (I) and observing that the number k is a factor of each equation. Thus if a system of n linear homogeneous equations has any solution distinct from the zero solution it has an infinite number of solutions.

CHAPTER XIX

PARTIAL FRACTIONS

202. Introduction. For various purposes it is convenient to

f(x)
(x)

§ 11, as the sum of

express a rational algebraic expression several fractions called partial fractions, which have the several factors of (x) as denominators and which have constants for numerators. If we write (x) = (x − α) (x − ẞ)... (x — v), we seek a means of determining constants A, B, N such that for every value of x

If the degree of f(x) is equal to or greater than that of (x),

where Q(x) is the quotient and f1 (x) the remainder from dividing f(x) by (x), and where the degree of f(x) is less than that of

(x). In what follows we shall assume that the degree of f(x) is less than that of (x). In problems where this is not the case one should carry out the long division indicated by (2) and apply the principle developed in this chapter to the expression corresponding to the last term in (2).

203. Development when (x) = 0 has no multiple roots. Let us consider the particular case

We indicate the development required in form (1) of the last

where A, B, and C represent constants which we are to determine if possible. The question arises immediately, Are we at liberty to make this assumption? Are we not assuming the essence of what we wish to prove, i.e. the form of the expansion? To this we may answer, We have written the expansion in form (1) tentatively. We have not proved it and are not certain of its validity. If, however, we are able to find numerical values of A, B, and C which satisfy (1), we can then write down the actual development of the fraction in the form of an identity.

If, on the other hand, we can show that no such numbers A, B, C satisfying (1) exist, then the development is not possible. Clear (1) of fractions,

x+1= A(x − 2) (x − 3) + B(x − 1) (x − 3) + C (x − 1) (x − 2)

= (A + B + C) x2 − (5 A + 4 B + 3 C) x + 6 A +3 B+ 2 C.

Since we seek values of A, B, and C for which (1) is identically true for all values of x, equate coefficients of like powers of x in the last equation (Corollary II, p. 174). We obtain

As a check we might clear of fractions and simplify. If equations (2), (3), and (4) had been incompatible, we should have concluded that we could not develop the fraction in form (1).

We assume now for the general case

Þ(x) = (x − α) (x − B) · · · (x − v),

and that the roots a, ß, v are all distinct from each other. Let us consider the expression

where A, B, ..., N are constants. Let us assume for the moment

f(x)
φ (α)

the possibility of expressing in terms of these partial frac

tions. We shall now attempt to determine actual values A, B, N which satisfy such an identity. If we multiply both sides of the identity by

we obtain

f(x) = A (x − ẞ) · · · (x − v) + B (x − x) · · · (x − v) + • • •

f(x) is of degree not greater than n − 1, and consequently when written in the form of (1), p. 166, has not more than n terms. If we multiply out the right-hand member and collect powers of x, we have an expression in x of degree n 1. By Corollary II, p. 174, this expression will be an identity if we can determine values of A, B, IN which make the coefficients of x on both sides of the equation equal to each other. Hence we equate coefficients of like powers of x and obtain n equations linear in A, B,, N which we can treat as variables. These equations have in general one and only one solution which we can easily determine. The values of A, B, N obtained by solving these equations we can substitute for the numerators of the partial fractions in (6). After making this substitution we can actually clear of fractions the right-hand member of (6) and check our work by showing its identity with the left-hand member.

There is no general criterion that we have applied to (6) to determine whether the n linear equations obtained by equating coefficients of x have any solution or not. Hence in this general