Taking the same process as in the preceding example,

SECTION 4.-Integration of some Particular Forms of Linear Differential Equations with Variable Coefficients.

459.] The linear differential equation of the following form admits of being reduced to one with constant coefficients by means of a change of variable, and therefore its integral may be determined by one or other of the preceding methods.

dry

(a + bx)" +11 (a + bx)n−1d"−ly

dxn−1 + 12 (a + bx)"−2.

n-2 d

dan-2

+..

+An-1 (a + bx)

and I may at once remark that if the equation admits of integration when the right-hand member vanishes, it may also be integrated when the right-hand member is a function of x.

Let a + bx = z; then as x is equicrescent in (142), so will also z be; and therefore after the substitution the equation is

dy

d2 y

dx

dx2

and so on; thus x

dy day

dt' dt2

x2

may be expressed in terms of

...; and (144) will become a linear differential equation

with constant coefficients.

...

460.] The form of linear differential equation which I shall consider next is

where the a's and the b's are constant. Now let us suppose a particular integral of this equation to be of a more general form than any heretofore assumed, and to be the definite integral of the form

y =

= √

eux v du,

(146)

where u is a new variable independent of x, v is a function of u, and u1 and u。 are the limits of integration and are independent of x; and let us consider the result of the substitution of this quantity in the given differential equation; differentiating (146), d"y u1

dy = ["we" vdu, dy = ["we" vdu,... "" = ["w" ev du;

dx

ueux

dx2

u2 eux

and moreover let us substitute as follows:

dx"

Now as v is an undetermined function of u, let us assume that

And in consequence of this assumption (147) becomes

but in this expression u1 and u, are undetermined; they must however satisfy (148); and as there will in general be no relation between them, each separately must satisfy it: and therefore we must discover the roots of the equation

to

let us suppose them to be u, u1, u,... u; then if we take be the inferior limit in all cases, and the others in turn to be superior limits, we have the following k values of y, viz.

and from the form of the equation it is plain that the sum of these also satisfies the equations. If therefore it is possible to find n+1 such values of u, the resulting expression of the form (149) is the general integral of the given equation; in other cases it may be only a particular integral.

And I must observe that the definite integrals which enter into the final result generally do not admit of further reduction; and hence we infer that the integral of a differential equation of the form (145) is a transcendent of a higher order than any of the commonly tabulated functions.

461.] The following are examples of which the integral is expressed as a definite integral.

.. Voа1u +α,

du = √ w2 + b1u + bo

= ▲ log (u− a)+B log (u—ß),

if a and B are the roots of the denominators, and A and B are determinate constants dependent on ɑ1, α, b1, b。: so that from (148) we have

eux (u-a)^(u-ẞ) = 0,

and this equation is satisfied by u = a, u = ß, u =

∞; and

these definite integrals do not admit of further reduction.

which is satisfied by u3∞; therefore u3 + a3 = 0, if in the result a = ∞ ; and of this equation, if r is a primitive cube root of -1, the roots are

Con

sequently if the primitive roots of (—1)"+ are 1, r, r2, ... r", and a is a quantity which, in the result, is infinite, the roots of this equation area, ar, ar2, ... ar";

Ex. 5. As the last example of this method let us take equation (260) in Art. 425, which is equivalent to Riccati's equation, and exhibit the function, which satisfies it, in the form of a definite integral. The equation may be put into the form