Hence, therefore, the repetition of a pair of imaginary roots a+b√-1 changes also the two arbitrary constants of the ordinary real solution into polynomials, each of which involves a number of constants equal to the number of times that the imaginary pair presents itself.

Ex. Given dy + 2n2 d'y + n'y = 0.

Assuming y= Cem, the auxiliary equation is

whence m has two pairs of roots of the form n√(− 1). For one such pair the form of solution would be

y = A cos nx + B sin nx.

For the actual case it therefore is

y = (A ̧ + Â ̧x) cos nx + (B2+ B2x) sin nx.

9. The above, which is the ordinary method of investigating the form of the complete solution when the auxiliary equation involves equal roots, rests on the assumption that a law of continuity connects the form of solution when roots are equal with the form of solution when the roots are unequal. Now, though it is perfectly true that such a law does exist, its assumption without proof of that existence must be regarded as opposed to the requirements of a strict logic. In all legitimate applications of the Differential Calculus it is with a limit that we are directly concerned. Here it is with something which exists, and which admits of being determined independently of the notion of a limit.

day dy

Thus if we take as an example -2- +y=0, in which dx2 dx

the auxiliary equation m2-2m+1=0 shews that the values of m are each equal to 1, we are entitled to assume as a particular solution

y = Ce

Let us now substitute this value of y in the given equation regarding C as variable, and inquire whether it admits of any more general determination than it has received above. On substitution we find simply

whence C=A+ Bx. Thus while the correctness of the solution furnished by the assumption of continuity is established, it is made manifest that this assumption is not indispensable.

We shall endeavour to establish upon other grounds the theory of these cases of failure in a future Chapter. Meanwhile it may be desirable to shew that the form (16) actually satisfies the differential equation when r values of m are equal.

s being an integer less than r. From the theorem for it easily follows that the result will be of the form

dm (uv) dx"

1.2

in which f(m) represents the first member of (11). But that equation having by hypothesis r equal roots, we know by the theory of equations that

f(m)=0, f'(m)=0,... f'(m) = 0,

are simultaneously true. Thus the differential equation is satisfied. And being satisfied for the particular value of y in question it is satisfied by (16), which is the sum of all such values.

10. The results of the previous investigation may be summed up in the following rule.

RULE. The coefficients being constant and the second member 0, form an auxiliary equation by assuming y= Cem2, and determine the values of m. Then the complete value of y will be expressed by a series of terms characterized as follows, viz. For each real distinct value of m there will exist a term CeTM2; for each pair of imaginary values a± b√(− 1), a term

each of the coefficients A, B, C being an arbitrary constant if the corresponding root occur only once, but a polynomial of the (r− 1)th degree with arbitrary constant coefficients, if the root occur r times.

whence it will be found that the values of m are

0, 1, 1, -1±√(−1).

The complete primitive therefore is

y=C+(C1 + C2x) e* + C1e cos x + Ce sin x.

11. To solve the linear equation with constant coefficients when its second member is not equal to 0.

The usual mode of solution is 1st to determine the complete value of y on the hypothesis that the second member is 0; 2ndly, to substitute its expression in the given equation regarding the arbitrary constants as variable parameters; 3rdly, to determine those parameters so as to satisfy the equation given.

Supposing the given equation to be of the nth degree, n parameters will be employed. These may evidently be subjected to any n-1 arbitrary conditions. Now that system of conditions which renders the discovery of the remaining relation (involved in the condition that the given differential

equation shall be satisfied) the most easy, is that which demands that the formal expression of the n-1 differential coefficients

shall, like the formal expression of y, be the same in the system in which c1, C.,,... C, represent variable parameters, as in the system in which they represent arbitrary constants.

n

The above method is commonly called the method of the variation of parameters. It is, as we shall hereafter see, far from being the easiest mode of solving the class of equations under consideration; but it is interesting as being probably the first general method discovered, and still more so from its containing an application of a principle successfully employed in higher problems.

Were the second member 0, the solution would be

y = c1 cos nx + c, sin nx.......

1

......

..(a).

Assume this then to be the form of the solution of the equa

dy

dx

tion given, c1, c, being variable parameters, but such that shall also retain the same form as if they were constant, viz.

This then is the condition which must accompany (a).

...(c).

Now differentiating (6) and regarding c1, c, as variable, we

have

and this equation, in combination with (c), gives

Lastly, substituting these values in (a) and reducing, we

and regarding the first term as a vanishing fraction when n=a, we find

Or we might proceed thus. Differentiating twice the equation