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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. (a + bx)"" ? + A, (a + bx)"–
+ A,(a + bx)"-29
... + An–1(a+bx) ai + Any = 0; (142) 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
on those dy and so on; thus x
... may be expressed in terms of
dx' “ dx2' di' dt? ";
...; and (144) will become a linear differential equation with constant coefficients.
so that by the methods of the preceding section
2 m y = C, X+Cgx-+
460.] The form of linear differential equation which I shall consider next is
... +(a,+1 (do + box)y = 0, (145) 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 = | eum v du,
(146) where u is a new variable independent of x, v is a function of u, and w, 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), dy pun d2u un
day -puu = | ueut v du,
u" elhal v du;
And in consequence of this assumption (147) becomes
but in this expression u, 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
eur do = 0; let us suppose them to be Up, U7, U9, ... Uz; then if we take u, to 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.
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
ce nu am – * xy = 0;
:. Sudu = n log (u? – 6°); and (148) becomes eux (u? — 62)" = 0;
ii U = -00, = +b = -b;