CHAPTER IV.

INTEGRATION OF DIFFERENTIAL EQUATIONS.

SECT. 1. Linear Equations with constant coefficients.

THESE form the largest class of Differential Equations which are integrable by one method, and they are of great importance, as many of the equations which are met with in the application of the Calculus to physics are either in this shape or may be reduced to it.

be the general form of a linear differential equation with constant coefficients; A1, A2... A, being constants, and X being any function of x. On separating the symbols of operation from those of quantity this becomes

d n-1

- =
A,

f

d

+A, + &c. ... + 4. } y = X or ƒ (1). y = X, (2)

dx

as we may write it for shortness. Now by the theorem given in Ex. 5 of Chap. xv. of the Differential Calculus,

d

the complex operation ƒ (~)
f is equivalent to

dx

a1, a2... a, being the roots of the equation f(x) = 0......(3).

Hence performing on both sides of (2) the inverse pro

The result of this transformation is different according

to the nature of the roots of (3).

1st. Let all the roots be unequal; then by the theorem given in Ex. 6. Chap. xv. of the Differential Calculus, the equation (4) becomes

where N1 =

1

X+

-1

(a1

(aa) (a, aз) ... (α, - a2)

and similarly for the other coefficients.

But by the theorem in Ex. 11 of the same Chapter,

A similar transformation being made of the other terms, we find

It is to be observed that each of the signs of integration would give rise to an arbitrary constant; and that this must be added in each of the terms when the integrations are effected. The value of y would then appear under the form

y = N1 €a1* ( [dx € ̄a‚a X + C1) + N2 ea2* ( [dx € ̄a2* X+C2) + &c.

2

+N2€an* ( ƒdx € ̃ ̄a‚a X + C„) ................ (7).

C1, C2... C, being the arbitrary constants.

.....

The functions Cear which arise in the integration are called complementary functions.

2nd. Let r of the roots of the equation (4) be equal

to a.

Then by the Theory of the decomposition of partial

fractions we know that the factor (-a) will give rise

to a series of r terms in (5) of the form

or, introducing the arbitrary constants which arise from the

There are in all exactly n arbitrary constants as there ought to be.

3rd. Let there be a pair of impossible roots, which must be of the form

a +(-) and a -
-(-) B;

then the coefficients of the corresponding terms in (6) are

{ (−)

And as ea+(-)ẞ} x = eax {cos ẞx + (-)1 sin ßæ},

and e{a-(-)ẞ} x = eax {cos ẞx − (−) sin ẞx}. The sum of the two corresponding terms in (6) is 2x (A cos ẞx + B sin ßx) fdx (e-ax cos ẞx. X) A2 + B2

+ 2ea* (A sin ẞæ – B cos ßx) fdx (e-aa sin ßx . X)

A2 + B2

This may be put under a simpler form, for if

(9).

-Fx

2 €* {cos (ẞx-0) ƒ d x (¤ ̄ cos ßæ X ) + sin (ẞ x − 0) ƒ d x ( € ̃a2 sin ßæ X )}

may evidently be put under the form

ea (C cos ẞx+ C' sin ẞx) = Ceаx cos (ẞx + a).

(10).

(11)

If there be a number of equal pairs of impossible roots in the equation (3), the general expression for the value of y becomes so complicated as to be of little use, and it is therefore unnecessary to insert it here.

The preceding process may frequently be simplified in its application to particular cases, by means of the following considerations. The inverse operations are always reduced

to the sum of several of the form

be 1 or any positive integer.

d

dx

a)

+ a

X where may

Now this operation will have a different effect according as it is expanded in ascending

inasmuch as in the one case it will involve integrals, while

in the other it will involve differentials only. But a simple relation connects the two, for

* (C1 + C1 x + C2 x2 + &c. + C‚...‚ x2-1), and the former term

d

-ax

= €

-T

X, being expanded in ascending powers of

d

dx

will give rise to a series of differentials which are always easily found, and which, when X is a rational and integral function of a of n dimensions, always breaks off at the (n + 1)th term. But since each factor of the form

da

gives rise to a separate complementary function, while X is operated on by all in succession, it is sufficient to expand

{()

-1

in descending powers of X, without splitting it

into its binomial factors, and then to add the complementary functions corresponding to each of these factors.

If the function be of the form e, the result of

mx

dx

+ &c.}ems,

and then operate one with each term separately, we find,

{A + Bm + Cm2 + Dm3 + &c.} em* = ƒ (m) e12.