4. If a factor, such as x2+ax+b, which is not resolvable into real linear factors occur in the denominator, the form of the correspond

the proper assumption for the form in partial fractions would be

where A, B, and C can be found according to the preceding methods, and on reduction to a common denominator we can, by equating coefficients of like powers in the two numerators, find the remaining letters D, E, F, G, H, K. Variations upon these methods will suggest themselves to the student.

13. If y = A sin mx + B cos mx, prove that

X

dx

x

dy
X- + 2.
dx

dya2y.

dx

24. If y

1
1+x+x2+x039

show that

dry is

dxn

(-1)"n! sin"+10 (sin(n + 1)0 - cos(n + 1) + (sin + cos 0)-"-1}

Ꮎ

[MATH. TRIPOS.

nedny

= a2y, prove that

=

(1 - x2) αy (2n + 1)x2

dx"+2

dx"

du+ly

- (n2 + a2) d′′y

0.

dx"

27. If y = sin(m sin-1x), show that (1 − x2).

28. If y = A(x + √x2 + a2)” + B(x+ √x2 + a2)−”,

dy X- m2y, dx

The last equation is to be found by substituting the series

y

in equation (i.) and equating the coefficients of a” to zero. 30. If sin(m sin ̄1x)=α + α1x + α„x2 + ..., prove

32. If (sin ̄1x)2 = A +а1x+а„x2 +ɑ«x«3 + show that

(n + 1)(n + 2)an+2= n2an•

33. If f(x) can be expanded in a series of positive integral powers of %, show that

eax pax.

d

ƒ (d)ea X = e^xf (x + 1) X,

where X represents any function of x.

36. Find the nth differential coefficient of ex {a2x2 - 2nax +n(n+1)}.

37. If u = sin nx + cos nx, show that

[I. C. S. EXAM.]

dru

(=n" {1+(-1)" sin 2nx.

[I. C. S. EXAM.]

1

38. If

ex

1

da

be differentiated i times, the denominator of the

result will be (e* − 1)'+1, and the sum of the coefficients of the several powers of e* in the numerator will be ( -- 1)‘1 . 2 . 3.....¿.

CHAPTER V.

EXPANSIONS.

102. The student will have already met with several expansions of given explicit functions in ascending integral powers of the independent variable; for example, those for (x+a)”, e*, log (1+x), tan-1x, sin x, cos x, which occur in ordinary Algebra and Trigonometry.

The principal methods of development in common use may be briefly classified as follows:

I. By purely Algebraical or Trigonometrical processes.
II. By Taylor's or Maclaurin's Theorems.

III. By Differentiation or Integration of a known series, or equivalent process.

IV. By the use of a differential equation.

These methods we proceed to explain and exemplify.

103. METHOD I. Algebraic and Trigonometrical Methods. Ex. 1. Find the first three terms of the expansion of log secx in ascending powers of x.