Chapter XII: Part 12

DIETRICH, CHRISTIAN WILHELM ERNST (1712-1774), German painter, was born at Weimar, where he was brought up early to the profession of art by his father Johann George, then painter of miniatures to the court of the duke. Having been sent to Dresden to perfect himself under the care of Alexander Thiele, he had the good fortune to finish in two hours, at the age of eighteen, a picture which attracted the attention of the king of Saxony. Augustus II. was so pleased with Dietrich's readiness of hand that he gave him means to study abroad, and visit in succession the chief cities of Italy and the Netherlands. There he learnt to copy and to imitate masters of the previous century with a versatility truly surprising. Winckelmann, to whom he had been recommended, did not hesitate to call him the Raphael of landscape. Yet in this branch of his practice he merely imitated Salvator Rosa and Everdingen. He was more successful in aping the style of Rembrandt, and numerous examples of this habit may be found in the galleries of St Petersburg, Vienna and Dresden. At Dresden, indeed, there are pictures acknowledged to be his, bearing the fictitious dates of 1636 and 1638, and the name of Rembrandt. Among Dietrich's cleverest reproductions we may account that of Ostade's manner in the "Itinerant Singers" at the National Gallery. His skill in catching the character of the later masters of Holland is shown in candlelight scenes, such as the "Squirrel and the Peep-Show" at St Petersburg, where we are easily reminded of Godfried Schalcken. Dietrich tried every branch of art except portraits, painting Italian and Dutch views alternately with Scripture scenes and still life. In 1741 he was appointed court painter to Augustus III. at Dresden, with an annual salary of 400 thalers (L60), conditional on the production of four cabinet pictures a year. This condition, no doubt, accounts for the presence of fifty-two of the master's panels and canvases in one of the rooms at the Dresden museum. Dietrich, though popular and probably the busiest artist of his time, never produced anything of his own; and his imitations are necessarily inferior to the originals which he affected to copy. His best work is certainly that which he gave to engravings. A collection of these at the British Museum, produced on the general lines of earlier men, such as Ostade and Rembrandt, reveal both spirit and skill. Dietrich, after his return from the Peninsula, generally signed himself "Dietericij," and with this signature most of his extant pictures are inscribed. He died at Dresden, after he had successively filled the important appointments of director of the school of painting at the Meissen porcelain factory and professor of the Dresden academy of arts.

DIETRICH OF BERN, the name given in German popular poetry to Theodoric the Great. The legendary history of Dietrich differs so widely from the life of Theodoric that it has been suggested that the two were originally unconnected. Medieval chroniclers, however, repeatedly asserted the identity of Dietrich and Theodoric, although the more critical noted the anachronisms involved in making Ermanaric (d. 376) and Attila (d. 453) contemporary with Theodoric (b. 455). That the legend is based on vague historical reminiscences is proved by the retention of the names of Theodoric (Thiuda-reiks, Dietrich) and his father Theudemir (Dietmar), by Dietrich's connexion with Bern (Verona) and Raben (Ravenna). Something of the Gothic king's character descended to Dietrich, familiarly called the Berner, the favourite of German medieval saga heroes, although his story did not leave the same mark on later German literature as did that of the Nibelungs. The cycle of songs connected with his name in South Germany is partially preserved in the Heldenbuch (q.v.) in _Dietrich's Flucht_, the _Rabenschlacht_ and _Alpharts Tod_; but it was reserved for an Icelandic author, writing in Norway in the 13th century, to compile, with many romantic additions, a consecutive account of Dietrich. In this Norse prose redaction, known as the _Vilkina Saga_, or more correctly the _Thidrekssaga_, is incorporated much extraneous matter from the Nibelungen and Wayland legends, in fact practically the whole of south German heroic tradition.

There are traces of a form of the Dietrich legend in which he was represented as starting out from Byzantium, in accordance with historical tradition, for his conquest of Italy. But this early disappeared, and was superseded by the existing legend, in which, perhaps by an "epic fusion" with his father Theudemir, he was associated with Attila, and then by an easy transition with Ermanaric. Dietrich was driven from his kingdom of Bern by his uncle Ermanaric. After years of exile at the court of Attila he returned with a Hunnish army to Italy, and defeated Ermanaric in the Rabenschlacht, or battle of Ravenna. Attila's two sons, with Dietrich's brother, fell in the fight, and Dietrich returned to Attila's court to answer for the death of the young princes. This very improbable renunciation of the advantages of his victory suggests that in the original version of the story the Rabenschlacht was a defeat. In the poem of _Ermenrichs Tod_ he is represented as slaying Ermanaric, as in fact Theodoric slew Odoacer. "Otacher" replaces Ermanaric as his adversary in the _Hildebrandslied_, which relates how thirty years after the earlier attempt he reconquered his Lombard kingdom. Dietrich's long residence at Attila's court represents the youth and early manhood of Theodoric spent at the imperial court and fighting in the Balkan peninsula, and, in accordance with epic custom, the period of exile was adorned with war-like exploits, with fights with dragons and giants, most of which had no essential connexion with the cycle. The romantic poems of _Konig Laurin_, _Sigenot_, _Eckenlied_ and _Virginal_ are based largely on local traditions originally independent of Dietrich. The court of Attila (Etzel) was a ready bridge to the Nibelungen legend. In the final catastrophe he was at length compelled, after steadily holding aloof from the combat, to avenge the slaughter of his Amelungs by the Burgundians, and delivered Hagen bound into the hands of Kriemhild. The flame breath which anger induced from him shows the influence of pure myth, but the tales of his demonic origin and of his being carried off by the devil in the shape of a black horse may safely be put down to the clerical hostility to Theodoric's Arianism.

Generally speaking, Dietrich of Bern was the wise and just monarch as opposed to Ermanaric, the typical tyrant of Germanic legend. He was invariably represented as slow of provocation and a friend of peace, but once roused to battle not even Siegfried could withstand his onslaught. But probably Dietrich's fight with Siegfried in Kriemhild's rose garden at Worms is a late addition to the Rosengarten myth. The chief heroes of the Dietrich cycle are his tutor and companion in arms, Hildebrand (see HILDEBRAND, lay of), with his nephews the Wolfings Alphart and Wolfhart; Wittich, who renounced his allegiance to Dietrich and slew the sons of Attila; Heime and Biterolf.

The contents of the poems dealing with the Dietrich cycle are
summarized by Uhland in _Schriften zur Geschichte der Dichtung und
Sage_ (Stuttgart, 1873). The _Thidrekssaga_ (ed. C. Unger,
Christiania, 1853) is translated into German by F. H. v. der Hagen in
_Altdeutsche und altnordische Heldensagen_ (vols. i. and ii. 3rd ed.,
Breslau, 1872). A summary of it forms the concluding chapter of T.
Hodgkin's _Theodoric the Goth_ (1891). The variations in the Dietrich
legend in the Latin historians, in Old and Middle High German
literature, and in the northern saga, can be studied in W. Grimm's
_Deutsche Heldensage_ (2nd ed., Berlin, 1867). There is a good account
in English in F. E. Sandbach's _Heroic Saga-cycle of Dietrich of Bern_
(1906), forming No. 15 of Alfred Nutt's _Popular Studies in
Mythology_, and another in M. Bentinck Smith's translation of Dr O. L.
Jiriczek's _Deutsche Heldensage_ (_Northern Legends_, London, 1902).
For modern German authorities and commentators see B. Symons,
"Deutsche Heldensage" in H. Paul's _Grd. d. german. Phil._
(Strassburg, new ed., 1905); also Goedeke, _Geschichte der deutschen
Dichtung_ (i. 241-246).

DIEZ, FRIEDRICH CHRISTIAN (1794-1876), German philologist, was born at Giessen, in Hesse-Darmstadt, on the 15th of March 1794. He was educated first at the gymnasium and then at the university of his native town. There he studied classics under Friedrich Gottlieb Welcker (1784-1868) who had just returned from a two years' residence in Italy to fill the chair of archaeology and Greek literature. It was Welcker who kindled in him a love of Italian poetry, and thus gave the first bent to his genius. In 1813 he joined the Hesse corps as a volunteer and served in the French campaign. Next year he returned to his books, and this short taste of military service was the only break in a long and uneventful life of literary labours. By his parents' desire he applied himself for a short time to law, but a visit to Goethe in 1818 gave a new direction to his studies, and determined his future career. Goethe had been reading Raynouard's _Selections from the Romance Poets_, and advised the young scholar to explore the rich mine of Provencal literature which the French savant had opened up. This advice was eagerly followed, and henceforth Diez devoted himself to Romance literature. He thus became the founder of Romance philology. After supporting himself for some years by private teaching, he removed in 1822 to Bonn, where he held the position of privatdocent. In 1823 he published his first work, _An Introduction to Romance Poetry_; in the following year appeared _The Poetry of the Troubadours_, and in 1829 _The Lives and Works of the Troubadours_. In 1830 he was called to the chair of modern literature. The rest of his life was mainly occupied with the composition of the two great works on which his fame rests, the _Grammar of the Romance Languages_ (1836-1844), and the _Lexicon of the Romance Languages--Italian, Spanish and French_ (1853); in these two works Diez did for the Romance group of languages what Jacob Grimm did for the Teutonic family. He died at Bonn on the 29th of May 1876.

The earliest French philologists, such as Perion and Henri Estienne,
had sought to discover the origin of French in Greek and even in
Hebrew. For more than a century Menage's _Etymological Dictionary_
held the field without a rival. Considering the time at which it was
written (1650), it was a meritorious work, but philology was then in
the empirical stage, and many of Menage's derivations (such as that of
"rat" from the Latin "mus," or of "haricot" from "faba") have since
become bywords among philologists. A great advance was made by
Raynouard, who by his critical editions of the works of the
Troubadours, published in the first years of the 19th century, laid
the foundations on which Diez afterwards built. The difference between
Diez's method and that of his predecessors is well stated by him in
the preface to his dictionary. In sum it is the difference between
science and guess-work. The scientific method is to follow implicitly
the discovered principles and rules of phonology, and not to swerve a
foot's breadth from them unless plain, actual exceptions shall justify
it; to follow the genius of the language, and by cross-questioning to
elicit its secrets; to gauge each letter and estimate the value which
attaches to it in each position; and lastly to possess the true
philosophic spirit which is prepared to welcome any new fact, though
it may modify or upset the most cherished theory. Such is the
historical method which Diez pursues in his grammar and dictionary. To
collect and arrange facts is, as he tells us, the sole secret of his
success, and he adds in other words the famous apophthegm of Newton,
"hypotheses non fingo." The introduction to the grammar consists of
two parts:--the first discusses the Latin, Greek and Teutonic elements
common to the Romance languages; the second treats of the six dialects
separately, their origin and the elements peculiar to each. The
grammar itself is divided into four books, on phonology, on flexion,
on the formation of words by composition and derivation, and on
syntax.

His dictionary is divided into two parts. The first contains words
common to two at least of the three principal groups of
Romance:--Italian, Spanish and Portuguese, and Provencal and French.
The Italian, as nearest the original, is placed at the head of each
article. The second part treats of words peculiar to one group. There
is no separate glossary of Wallachian.

Of the introduction to the grammar there is an English translation by
C. B. Cayley. The dictionary has been published in a remodelled form
for English readers by T. C. Donkin.

DIEZ, a town of Germany, in the Prussian province of Hesse-Nassau, romantically situated in the deep valley of the Lahn, here crossed by an old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500. It is overlooked by a former castle of the counts of Nassau-Dillenburg, now a prison. Close by, on an eminence above the river, lies the castle of Oranienstein, formerly a Benedictine nunnery and now a cadet school, with beautiful gardens. There are a Roman Catholic and two Evangelical churches. The new part of the town is well built and contains numerous pretty villa residences. In addition to extensive iron-works there are sawmills and tanneries. In the vicinity are Fachingen, celebrated for its mineral waters, and the majestic castle of Schaumburg belonging to the prince of Waldeck-Pyrmont.

DIFFERENCES, CALCULUS OF (_Theory of Finite Differences_), that branch of mathematics which deals with the successive differences of the terms of a series.

1. The most important of the cases to which mathematical methods can be applied are those in which the terms of the series are the values, taken at stated intervals (regular or irregular), of a continuously varying quantity. In these cases the formulae of finite differences enable certain quantities, whose exact value depends on the law of variation (i.e. the law which governs the relative magnitude of these terms) to be calculated, often with great accuracy, from the given terms of the series, without explicit reference to the law of variation itself. The methods used may be extended to cases where the series is a double series (series of double entry), i.e. where the value of each term depends on the values of a pair of other quantities.

2. The _first differences_ of a series are obtained by subtracting from each term the term immediately preceding it. If these are treated as terms of a new series, the first differences of this series are the _second differences_ of the original series; and so on. The successive differences are also called _differences of the first, second, ... order_. The differences of successive orders are most conveniently arranged in successive columns of a table thus:--

+-----+----------+-----------+-----------------+----------------------+
|Term.| 1st Diff.| 2nd Diff. | 3rd Diff. | 4th Diff. |
+-----+----------+-----------+-----------------+----------------------+
| | | | | |
| a | | | | |
| | b - a | | | |
| b | | c - 2b +a | | |
| | c - b | | d - 3c + 3b - a | |
| c | | d - 2c +b | | e - 4d + 6c - 4b + a |
| | d - c | | e - 3d + 3c - b | |
| d | | e - 2d +c | | |
| | e - d | | | |
| e | | | | |
+-----+----------+-----------+-----------------+----------------------+

_Algebra of Differences and Sums._

3. The formal relations between the terms of the series and the
differences may be seen by comparing the arrangements (A) and (B) in
fig. 1. In (A) the various terms and differences are the same as in S
2, but placed differently. In (B) we take a new series of terms
[alpha], [beta], [gamma], [delta], commencing with the same term
[alpha], and take the successive sums of pairs of terms, instead of
the successive differences, but place them to the left instead of to
the right. It will be seen, in the first place, that the successive
terms in (A), reading downwards to the right, and the successive terms
in (B), reading downwards to the left, consist each of a series of
terms whose coefficients follow the binomial law; i.e. the
coefficients in b - a, c - 2b + a, d - 3c + 3b - a, ... and in [alpha]
+ [beta], [alpha] + 2[beta] + [gamma], [alpha] + 3[beta] + 3[gamma] +
[delta], ... are respectively the same as in y - x, (y - x)^2, (y -
x)^3, ... and in x + y, (x + y)^2, (x + y)^3,.... In the second place,
it will be seen that the relations between the various terms in (A)
are identical with the relations between the similarly placed terms in
(B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] +
[gamma] and [alpha] + [beta], just as c - b is the difference of c and
b: and d - c is the sum of c - b and d - 2c + b, just as [beta] +
2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] +
[delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as
being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all
corresponding terms in the two diagrams will be the same.

Thus we obtain the two principal formulae connecting terms and
differences. If we provisionally describe b - a, c - 2b + a, ... as
the first, second, ... differences of the particular term a (S 7),
then (i.) the nth difference of a is

n.n - 1
l - nk + ... + (-1)^(n-2) ------- c + (-1)^(n-1) nb + (-1)^n a,
1.2

where l, k ... are the (n + 1)th, nth, ... terms of the series a, b,
c, ...; the coefficients being those of the terms in the expansion of
(y -x)^n: and (ii.) the (n + 1)th term of the series, i.e. the nth
term after a, is

n.n - 1
a + n[beta] + ------- [gamma] + ...
1.2

where [beta], [gamma], ... are the first, second, ... differences of
a; the coefficients being those of the terms in the expansion of (x +
y)^n.

4. Now suppose we treat the terms a, b, c, ... as being themselves the
first differences of another series. Then, if the first term of this
series is N, the subsequent terms are N + a, N + a + b, N + a + b + c,
...; i.e. the difference between the (n + 1)th term and the first term
is the sum of the first n terms of the original series. The term N, in
the diagram (A), will come above and to the left of a; and we see, by
(ii.) of S 3, that the sum of the first n terms of the original series
is

/ n.n - 1 \ n.n - 1 n.n - 1.n - 2
( N + na + ------- [beta] + ...) - N = na + ------- [beta] + ------------- [gamma] + ...
\ 1.2 / 1.2 1 . 2 . 3

5. As an example, take the arithmetical series

a, a + p, a + 2p, ...

The first differences are p, p, p, ... and the differences of any
higher order are zero. Hence, by (ii.) of S 3, the (n + 1)th term is a
+ np, and, by S 4, the sum of the first n terms is na + 1/2n(n - 1)p =
1/2n{2a + (n - 1)p}.

6 As another example, take the series 1, 8, 27, ... the terms of which
are the cubes of 1, 2, 3, ... The first, second and third differences
of the first term are 7, 12 and 6, and it may be shown (S 14 (i.))
that all differences of a higher order are zero. Hence the sum of the
first n terms is

n.n - 1 n.n - 1.n - 2 n.n - 1.n - 2.n - 3
n + 7 ------- + 12 ------------- + 6 ------------------- =
1.2 1.2.3 1.2.3.4

1/4n^4 + 1/2n^3 + 1/4n^2 = {1/2n(n + 1)}^2.

7. In S 3 we have described b - a, c - 2b + a, ... as the first,
second, ... differences of a. This ascription of the differences to
particular terms of the series is quite arbitrary. If we read the
differences in the table of S 2 upwards to the right instead of
downwards to the right, we might describe e - d, e - 2d + c, ... as
the first, second, ... differences of e. On the other hand, the term
of greatest weight in c -2b + a, i.e. the term which has the
numerically greatest coefficient, is b, and therefore c - 2b + a might
properly be regarded as the second difference of b, and similarly e -
4d + 6c - 4b + a might be regarded as the fourth difference of c.
These three methods of regarding the differences lead to three
different systems of notation, which are described in SS 9, 10 and 11.

_Notation of Differences and Sums._

8. It is convenient to denote the terms a, b, c, ... of the series by
u0, u1, u2, u3, ... If we merely have the terms of the series, un may
be regarded as meaning the (n + 1)th term. Usually, however, the terms
are the values of a quantity u, which is a function of another
quantity x, and the values of x, to which a, b, c, ... correspond,
proceed by a constant difference h. If x0 and u0 are a pair of
corresponding values of x and u, and if any other value x0 + mh of x
and the corresponding value of u are denoted by xm and um, then the
terms of the series will be ... u_(n-2), u_(n-1), u_n, u_(n+1),
u_(n+2) ..., corresponding to values Of x denoted by ... x_(n-2),
x_(n-1), x_n, x_(n+1), x_(n+2)....

9. In the _advancing-difference notation_ u_(n+1) - u_n is denoted by
[Delta]un. The differences [Delta]u0, [Delta]u1, [Delta]u2 ... may
then be regarded as values of a function [Delta]u corresponding to
values of x proceeding by constant difference h; and therefore
[Delta]u_(n+1) -[Delta]u_n denoted by [Delta][Delta]u_n, or, more
briefly, [Delta]^2u_n; and so on. Hence the table of differences in S
2, with the corresponding values of x and of u placed opposite each
other in the ordinary manner of mathematical tables, becomes

+---------+---------+----------------+-----------------+-----------------+----------------------+
| x | u | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
+---------+---------+----------------+-----------------+-----------------+----------------------+
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| | | | | | |
| x_(n-2) | u_(n-2) | | [Delta]^2u_(n-3)| | [Delta]^4u_(n-4) ... |
| | | [Delta]u_(n-2) | | [Delta]^3u_(n-3)| |
| x_(n-1) | u_(n-1) | | [Delta]^2u_(n-2)| | [Delta]^4u_(n-3) ... |
| | | [Delta]u_(n-1) | | [Delta]^3u_(n-2)| |
| xn | u_n | | [Delta]^2u_(n-1)| | [Delta]^4u_(n-2) ... |
| | | [Delta]u_n | | [Delta]^3u_(n-1)| |
| x_(n+1) | u_(n+1) | | [Delta]^2u_n | | [Delta]^4u_(n-1) ... |
| | | [Delta]u_(n+1) | | [Delta]^3u_n | |
| x_(n+2) | u_(n+2) | | [Delta]^2u_(n+1)| | [Delta]^4u_n ... |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
+---------+---------+----------------+-----------------+-----------------+----------------------+

The terms of the series of which ... u_(n-1), u_n, u_(n+1), ... are
the first differences are denoted by [Sigma]u, with proper suffixes,
so that this series is ... [Sigma]u_(n-1), [Sigma]u_n,
[Sigma]u_(n+1).... The suffixes are chosen so that we may have
[Delta][Sigma]un = un, whatever n may be; and therefore (S 4)
[Sigma]un may be regarded as being the sum of the terms of the series
up to and including un-1. Thus if we write [Sigma]u_(n-1) = C + un-2,
where C is any constant, we shall have

[Sigma]u_n = [Sigma]u_(n-1) + [Delta][Sigma]u_(n-1) = C + u_(n-2) + u_(n-1),
[Sigma]u_(n+1) = C + u_(n-2) + u_(n-1) + u_n,

and so on. This is true whatever C may be, so that the knowledge of
... u_n-1, u_n, ... gives us no knowledge of the exact value of
[Sigma]u_n; in other words, C is an arbitrary constant, the value of
which must be supposed to be the same throughout any operations in
which we are concerned with values of [Sigma]_u corresponding to
different suffixes.

There is another symbol E, used in conjunction with u to denote the
next term in the series. Thus Eun means u_(n+1), so that Eun = u_n +
[Delta]u_n.

10. Corresponding to the advancing-difference notation there is a
_receding-difference_ notation, in which u_(n+1) - u_n is regarded as
a difference of u_(n+1), and may be denoted by [Delta]'u_(n+1), and
similarly u_(n+1) - 2u_n + u_(n-1) may be denoted by [Delta]'^2u_(n+1).
This notation is only required for certain special purposes, and the
usage is not settled (S 19 (ii.)).

11. The _central-difference_ notation depends on treating u_(n+1) -
2u_n -u_(n-1) as the second difference of un, and therefore as
corresponding to the value x_n; but there is no settled system of
notation. The following seems to be the most convenient. Since un is a
function of x_n, and the second difference u_(n+2) - 2u_(n+1) + u_n is
a function of x_(n+1), the first difference u_(n+1) - u_n must be
regarded as a function of x_(n+1/2), i.e. of 1/2{x_n + x_(n+1)}. We
therefore write u_(n+1) - u_n = [delta]u_(n+1/2), and each difference in
the table in S 9 will have the same suffix as the value of x in the
same horizontal line; or, if the difference is of an odd order, its
suffix will be the means of those of the two nearest values of x. This
is shown in the table below.

In this notation, instead of using the symbol E, we use a symbol [mu]
to denote the mean of two consecutive values of u, or of two
consecutive differences of the same order, the suffixes being assigned
on the same principle as in the case of the differences. Thus

[mu]u_(n+1/2) = 1/2{u_n + u_(n+1)}, [mu][delta]u_n = 1/2{[delta]u_(n-1/2)} + [delta]u_(n+1/2), &c.

If we take the means of the differences of odd order immediately above
and below the horizontal line through any value of x, these means,
with the differences of even order in that line, constitute the
_central differences_ of the corresponding value of u. Thus the table
of central differences is as follows, the values obtained as means
being placed in brackets to distinguish them from the actual
differences:--

+-------+-------+---------------------+----------------+----------------------+----------------------+
| x | u | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
+-------+-------+---------------------+----------------+----------------------+----------------------+
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
|x_(n-2)|u_(n-2)| {[mu][delta]u_(n-2)}|[delta]^2u_(n-2)|{[mu][delta]^3u_(n-2)}| [delta]^4u_(n-2) ... |
| | | [delta]u_(n-3/2) | | [delta]^3u_(n-3/2) | |
|x_(n-1)|u_(n-1)| {[mu][delta]u_(n-1)}|[delta]^2u_(n-1)|{[mu][delta]^3u_(n-1)}| [delta]^4u_(n-1) ... |
| | | [delta]u_(n-1/2) | | [delta]^3u_(n-2 | |
|x_n |u_n | ([mu][delta]u_n) |[delta]^2u_n | ([mu][delta]^3u_n) | [delta]^4u_n ... |
| | | [delta]u_(n+1/2) | | [delta]^3u_(n+1/2) | |
|x_(n+1)|u_(n+1)| {[mu][delta]u_(n+1)}|[delta]^2u_(n+1)|{[mu][delta]^3u_(n+1)}| [delta]^4u_(n+1) ... |
| | | [delta]u_(n+3/2) | | [delta]^3u_(n+3/2) | |
|x_(n+2)|u_(n+2)| {[mu][delta]u_(n+2)}|[delta]^2u_(n+2)|{[mu][delta]^3u_(n+2)}| [delta]^4u_(n+2) ... |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
+-------+-------+---------------------+----------------+----------------------+----------------------+

Similarly, by taking the means of consecutive values of u and also of
consecutive differences of even order, we should get a series of terms
and differences central to the intervals x_(n-2) to x_(n-1), x_(n-1)
to x_n, ....

The terms of the series of which the values of u are the first
differences are denoted by [sigma]u, with suffixes on the same
principle; the suffixes being chosen so that [delta][sigma]un shall be
equal to un. Thus, if

[sigma]u_(n-3/2) = C + u_(n-2),

then

[sigma]u_(n-1/2) = C + u_(n-2) + u_(n-1), [sigma]_(n+1/2)
= C + u_(n-2) + u_(n-1) + u_n, &c.,

and also

[mu][sigma]u_(n-1) = C + u_(n-2) + 1/2u_(n-1), [mu][sigma]u_n
= C + u_(n-2) + u_(n-1) + 1/2u_n, &c.,

C being an arbitrary constant which must remain the same throughout
any series of operations.

_Operators and Symbolic Methods._

12. There are two further stages in the use of the symbols [Delta],
[Sigma], [delta], [sigma], &c., which are not essential for elementary
treatment but lead to powerful methods of deduction.

(i.) Instead of treating [Delta]u as a function of x, so that
[Delta]u_n means ([Delta]u)_n, we may regard [Delta] as denoting an
_operation_ performed on u, and take [Delta]un as meaning [Delta].u_n.
This applies to the other symbols E, [delta], &c., whether taken
simply or in combination. Thus [Delta]Eu_n means that we first replace
un by un+1, and then replace this by u_(n+2) - u_(n+1).

(ii.) The operations [Delta], E, [delta], and [mu], whether performed
separately or in combination, or in combination also with numerical
multipliers and with the operation of differentiation denoted by D (:=
d/dx), follow the ordinary rules of algebra: e.g. [Delta](u_n + v_n) =
[Delta]u_n + [Delta]v_n, [Delta]Du_n = D[Delta]u_n, &c. Hence the
symbols can be separated from the functions on which the operations
are performed, and treated as if they were algebraical quantities. For
instance, we have

E.u_n = u_(n+1) = u_n + [Delta]u_n = 1.u_n + [Delta].u_n,

so that we may write E = 1 + [Delta], or [Delta] = E - 1. The first of
these is nothing more than a statement, in concise form, that if we
take two quantities, subtract the first from the second, and add the
result to the first, we get the second. This seems almost a truism.
But, if we deduce E^n = (1 + [Delta])^n, [Delta]^n = (E-1)^n, and
expand by the binomial theorem and then operate on u0, we get the
general formulae

n.n - 1
un = u0 + n[Delta]u0 + ------- [Delta]^2u0 + ... + [Delta]^nu0,
1.2
n.n - 1
[Delta]^nu0 = u_n - nu_(n-1) + ------- u_(n-2) + ... + (-1)^nu0,
1.2

which are identical with the formulae in (ii.) and (i.) of S 3.

(iii.) What has been said under (ii.) applies, with certain
reservations, to the operations [Sigma] and [sigma], and to the
operation which represents integration. The latter is sometimes
denoted by D^-1; and, since [Delta][Sigma]un = un, and
[delta][sigma]u_n = u_n, we might similarly replace [Sigma] and
[sigma] by [Delta]^-1 and [delta]^-1. These symbols can be combined
with [Delta], E, &c. according to the ordinary laws of algebra,
provided that proper account is taken of the arbitrary constants
introduced by the operations D^-1, [Delta]^-1, [delta]^-1.

_Applications to Algebraical Series._

13. _Summation of Series._--If ur, denotes the (r+1)th term of a
series, and if vr is a function of r such that [Delta]v_r = u_r for
all integral values of r, then the sum of the terms u_m, u_(m+1), ...
un is v_(n+1) -v_m. Thus the sum of a number of terms of a series may
often be found by inspection, in the same kind of way that an integral
is found.

14. _Rational Integral Functions._--(i.) If u_r is a rational integral
function of r of degree p, then [Delta]ur, is a rational integral
function of r of degree p-1.

(ii.) A particular case is that of a _factorial_, i.e. a product of
the form (r+a+1) (r+a+2) ... (r+b), each factor exceeding the
preceding factor by 1. We have

[Delta].(r+a+1) (r+a+2) ... (r+b) = (b-a).(r+a+2) ... (r+b),

whence, changing a into a-1,

[Sigma](r+a+1)(r+a+2) ... (r+b) = _const._ + (r+a)(r+a+1) ...
(r+b)/(b-a+1).

A similar method can be applied to the series whose (r+1)th term is of
the form 1/(r+a+1) (r+a+2) ... (r+b).

(iii.) Any rational integral function can be converted into the sum of
a number of factorials; and thus the sum of a series of which such a
function is the general term can be found. For example, it may be
shown in this way that the sum of the pth powers of the first n
natural numbers is a rational integral function of n of degree p+1,
the coefficient of n^p+1 being 1/(p+1).

15. _Difference-equations._--The summation of the series ... + u_(n+2)
+ u_(n-1) + u_n is a solution of the _difference-equation_ [Delta]v_n
= u_(n+1), which may also be written (E-1)v_n = u_(n+1). This is a
simple form of difference-equation. There are several forms which have
been investigated; a simple form, more general than the above, is the
_linear equation_ with _constant coefficients_--

v_(n+m) + a1v_(n+m-1) + a2v_(n+m-2) + ... + a_mv_n = N,

where a1, a2, ... am are constants, and N is a given function of n.
This may be written

(E^m + a1E^(m-1) + ... + a_m)v_n = N

or

(E-p1)(E-p2) ... (E-p_m)v_n = N.

The solution, if p1, p2, ... pm are all different, is vn = C1p1^n +
C2p2^n + ... + C_mp_m^n + V_n, where C1, C2 ... are constants, and v_n
= V_n is any one solution of the equation. The method of finding a
value for Vn depends on the form of N. Certain modifications are
required when two or more of the p's are equal.

It should be observed, in all cases of this kind, that, in describing
C1, C2 as "constants," it is meant that the value of any one, as C1,
is the same for all values of n occurring in the series. A "constant"
may, however, be a periodic function of n.

_Applications to Continuous Functions._

16. The cases of greatest practical importance are those in which u is
a continuous function of x. The terms u1, u2 ... of the series then
represent the successive values of u corresponding to x = x1, x2....
The important applications of the theory in these cases are to (i.)
relations between differences and differential coefficients, (ii.)
interpolation, or the determination of intermediate values of u, and
(iii.) relations between sums and integrals.

17. Starting from any pair of values x0 and u0, we may suppose the
interval h from x0 to x1 to be divided into q equal portions. If we
suppose the corresponding values of u to be obtained, and their
differences taken, the successive advancing differences of u0 being
denoted by dPu0, dP^2u0 ..., we have (S 3 (ii.))

q.q - 1
u1 = u0 + qdPu0 + ------- dP^2u0 + ....
1.2

When q is made indefinitely great, this (writing f(x) for u) becomes
Taylor's Theorem (INFINITESIMAL CALCULUS)

h^2
f(x + h) = f(x) + hf'(x) + --- f"(x) + ...,
1.2

which, expressed in terms of operators, is

h^2 h^3
E = 1 + hD + ---D^2 + ----- D^3 + ... = e^(hD).
1.2 1.2.3

This gives the relation between [Delta] and D. Also we have

2q.2q - 1
u2 = u0 + 2qdPu0 + --------- dP^2u0 + ...
1.2

3q.3q - 1
u3 = u0 + 3qdPu0 + --------- dP^2u0 + ...
1.2
. .
. .
. .

and, if p is any integer,

p.p - 1
u_(p/q) = u0 + pdPu0 + ------- dP^2u0 + ....
1.2

From these equations up/q could be expressed in terms of u0, u1, u2,
...; this is a particular case of interpolation (q.v.).

18. _Differences and Differential Coefficients._--The various formulae
are most quickly obtained by symbolical methods; i.e. by dealing with
the operators [Delta], E, D, ... as if they were algebraical
quantities. Thus the relation E = e^(hD) (S 17) gives

hD = log_e (1 + [Delta]) = [Delta] - 1/2[Delta]^2 + 1/3 [Delta]^3 ...

/du\
or h( -- ) = [Delta]u0 - 1/2[Delta]^2u0 + 1/3 [Delta]^3u0 ....
\dx/0

The formulae connecting central differences with differential
coefficients are based on the relations [mu] = cosh 1/2hD = 1/2(e^1/2hD
+ e^ -1/2hD), [delta] = 2 sinh 1/2hD - e^ 1/2hD - e^ -1/2hD, and
may be grouped as follows:--

u0 = u0 \
|
[mu][delta]u0 = (hD + 1/6 h^3D^3 + 1/120 h^5 D^5 + ...)u0 |
|
[delta]^2u0 = (h^2D^2 + 1/12 h^4 D^4 + 1/360 h^6 D^6 + ...)u0 >
|
[mu][delta]^3u0 = (h^3D^3 + 1/4 h^5 D^5 + ...)u0 |
|
[delta]^4 u0 = (h^4 D^4 + 1/6 h^6 D^6 + ...)u0 /

. . .
. . .
. . .

[mu]u_1/2 = (1 + 1/8 h^2D^2 + 1/384 h^4 D^4 + 1/46080 h^6 D^6 + ...)u_1/2 \
|
[delta]u_1/2 = (hD + 1/24 h^3D^3 + 1/1920 h^5 D^5 + ...)u_1/2 |
|
[mu][delta]^2u_1/2 = (h^2D^2 + 5/24 h^4 D^4 + 91/5760 h^6 D^6 + ...)u_1/2 >
|
[delta]^3u_1/2 = (h^3D^3 + 1/8 h^5 D^5 + ...)u_1/2 |
|
[mu][delta]^4 u_1/2 = (h^4 D^4 + 7/24 h^6 D^6 + ...)u_1/2 /

. . .
. . .
. . .

u0 = u0 \
|
hDu0 = ([mu][delta] - 1/6 [mu][delta]^3 + 1/30 [mu][delta]^5 - ...)u0 |
|
h^2D^2u0 = ([delta]^2 - 1/12 [delta]^4 + 1/90 [delta]^6 - ...)u0 >
|
h^3D^3u0 = ([mu][delta]^3 1/4 [mu][delta]^5 + ...)u0 |
|
h^4 D^4 u_0 = ([delta]^4 - 1/6 [delta]^6 + ...)u0 /

. . .
. . .
. . .

u_1/2 = ([mu] - 1/8 [mu][delta]^2 + 3/128 [mu][delta]^4 - 5/1024 [mu][delta]^6 + ...)u_1/2 \
|
hDu_1/2 = ([delta] - 1/24 [delta]^3 + 3/640 [delta]^5 - ...)u_1/2 |
|
h^2D^2u_1/2 = ([mu][delta]^2 - 5/24 [mu][delta]^ + 259/5760 [mu][delta]^6 - ...)u_1/2 >
|
h^3D^3u_1/2 = ([delta]^3 - 1/8 [delta]^5 + ...)u_1/2 |
|
h^4 D^4 u_1/2 = ([mu][delta]^4 - 7/24 [mu][delta]^6 + ...)u_1/2 /

. . .
. . .
. . .

When u is a rational integral function of x, each of the above series
is a terminating series. In other cases the series will be an infinite
one, and may be divergent; but it may be used for purposes of
approximation up to a certain point, and there will be a "remainder,"
the limits of whose magnitude will be determinate.

19. _Sums and Integrals._--The relation between a sum and an integral
is usually expressed by the _Euler-Maclaurin formula_. The principle
of this formula is that, if um and um+1, are ordinates of a curve,
distant h from one another, then for a first approximation to the area
of the curve between um and um+1 we have 1/2h(u_m + u_m+1), and the
difference between this and the true value of the area can be
expressed as the difference of two expressions, one of which is a
function of x_m, and the other is the same function of x_m+1. Denoting
these by [phi](x_m) and [phi](xm+1), we have

_ x_m+1
/
| udx = 1/2h(u_m + u_m+1) + [phi](x_m+1 ) - [phi](x_m).
_/x_m

Adding a series of similar expressions, we find

_ x_n
/
| udx = h{1/2u_m + u_m+1 + u_m+2 + ... + u_n-1 + 1/2u_n} + [phi](x_n) - [phi](x_m).
_/x_m

The function [phi](x) can be expressed in terms either of differential
coefficients of u or of advancing or central differences; thus there
are three formulae.

(i.) The Euler-Maclaurin formula, properly so called, (due
independently to Euler and Maclaurin) is

_ x_n
/ 1 du_n 1 d^3u_n 1 d^5 u_n
| udx = h.[mu][sigma]u_n - -- h^2 ---- + --- h^4 ------ - ----- h^6 ------- + ...
_/x_m 12 dx 720 dx^3 30240 dx^5

B1 du_n B2 d^3u_n B3 d^5u_n
= h.[mu][sigma]u_n - -- h2 ---- + -- h^4 ------ - -- h^6 ------ + ...,
2! dx 4! dx^3 6! dx^5

where B1, B2, B3 ... are _Bernoulli's numbers_.

(ii.) If we express differential coefficients in terms of advancing
differences, we get a theorem which is due to Laplace:--

_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) - 1/12 ([Delta]u_n - [Delta]u0) + 1/24 ( [Delta]^2u_n - [Delta]^2u0)
h _/x0

- 19/720 ([Delta]^3u_n - [Delta]^3u_0) + 3/160 ([Delta]^4 u_n - [Delta]^4 u0) - ...

For practical calculations this may more conveniently be written

_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - 1/2[Delta]^2u0 + 19/60 [Delta]^3u0 - ...)
h _/x0

+ 1/12 ([Delta]'u_n - 1/2[Delta]'^2u_n + 19/60 [Delta]'^3u_n - ...),

where accented differences denote that the values of u are read
backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in S
10) u_n - u_n-1.

(iii.) Expressed in terms of central differences this becomes

_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]^3u_n - ...
h _/x0
+ 1/12 [mu][delta]u0 - 11/720 [mu][delta]^3u0 + ...

/ 1 11 191 2497 \ / \
= [mu]([sigma] - -- [delta] + --- [delta]^3 - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0).
\ 12 720 60480 3628800 / \ /

(iv.) There are variants of these formulae, due to taking hum+1/2 as the
first approximation to the area of the curve between um and um+1; the
formulae involve the sum u_1/2 + u_3/2 + ... + u_n-1/2 := [sigma](u_n -
u0) (see MENSURATION).

20. The formulae in the last section can be obtained by symbolical
methods from the relation

_
1 / 1 1
- | udx = - D^1 u = --.u.
h _/ h hD

Thus for central differences, if we write [theta] := 1/2hD, we have [mu]
= cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and
the result in (iii.) corresponds to the formula

/ / 1 2 2.4 \
sinh [theta] = [theta] cosh [theta]/ (1 + - sinh^2[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...).
/ \ 3 3.5 3.5.7 /

REFERENCES.--There is no recent English work on the theory of finite
differences as a whole. G. Boole's _Finite Differences_ (1st ed.,
1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive
treatise, in which symbolical methods are employed very early. A. A.
Markoff's _Differenzenrechnung_ (German trans., 1896) contains general
formulae. (Both these works ignore central differences.) _Encycl. der
math. Wiss._ vol. i. pt. 2, pp. 919-935, may also be consulted. An
elementary treatment of the subject will be found in many text-books,
e.g. G. Chrystal's _Algebra_ (pt. 2, ch. xxxi.). A. W. Sunderland,
_Notes on Finite Differences_ (1885), is intended for actuarial
students. Various central-difference formulae with references are
given in _Proc. Lond. Math. Soc._ xxxi. pp. 449-488. For other
references see INTERPOLATION. (W. F. SH.)

DIFFERENTIAL EQUATION, in mathematics, a relation between one or more functions and their differential coefficients. The subject is treated here in two parts: (1) an elementary introduction dealing with the more commonly recognized types of differential equations which can be solved by rule; and (2) the general theory.

_Part I.--Elementary Introduction._

Of equations involving only one independent variable, x (known as
_ordinary_ differential equations), and one dependent variable, y, and
containing only the first differential coefficient dy/dx (and
therefore said to be of the first _order_), the simplest form is that
reducible to the type

dy/dx = f(x)/F(y),

leading to the result fF(y)dy - ff(x)dx = A, where A is an arbitrary
constant; this result is said to solve the differential equation, the
problem of evaluating the integrals belonging to the integral
calculus.

Another simple form is

dy/dx + yP = Q,

where P, Q are functions of x only; this is known as the linear
equation, since it contains y and dy/dx only to the first degree. If

fPdx = u, we clearly have

d /dy \
--(ye^u) =e^u ( -- + Py) = e^u Q,
dx \dx /

so that y = e^-u(fe^u Qdx + A) solves the equation, and is the only
possible solution, A being an arbitrary constant. The rule for the
solution of the linear equation is thus to multiply the equation by
e^u, where u = fPdx.

A third simple and important form is that denoted by

y = px + f(p),

where p is an abbreviation for dy/dx; this is known as Clairaut's
form. By differentiation in regard to x it gives

dp dp
p = p + x-- + f'(p)--,
dx dx

where

d
f'(p) = -- f(p);
dp

thus, either (i.) dp/dx = 0, that is, p is constant on the curve
satisfying the differential equation, which curve is thus any one of
the straight lines y = cx = f(c), where c is an arbitrary constant, or
else, (ii.) x + [f]'(p) = 0; if this latter hypothesis be taken, and p
be eliminated between x + f'(p) = 0 and y = px + f(p), a relation
connecting x and y, not containing an arbitrary constant, will be
found, which obviously represents the envelope of the straight lines y
= cx + f(c).

In general if a differential equation [phi](x, y, dy/dx) = 0 be
satisfied by any one of the curves F(x, y, c) = 0, where c is an
arbitrary constant, it is clear that the envelope of these curves,
when existent, must also satisfy the differential equation; for this
equation prescribes a relation connecting only the co-ordinates x, y
and the differential coefficient dy/dx, and these three quantities are
the same at any point of the envelope for the envelope and for the
particular curve of the family which there touches the envelope. The
relation expressing the equation of the envelope is called a
_singular_ solution of the differential equation, meaning an
_isolated_ solution, as not being one of a family of curves depending
upon an arbitrary parameter.

An extended form of Clairaut's equation expressed by

y = xF(p) + f(p)

may be similarly solved by first differentiating in regard to p, when
it reduces to a linear equation of which x is the dependent and p the
independent variable; from the integral of this linear equation, and
the original differential equation, the quantity p is then to be
eliminated.

Other types of solvable differential equations of the first order are
(1)

M dy/dx = N,

where M, N are homogeneous polynomials in x and y, of the same order;
by putting v = y/x and eliminating y, the equation becomes of the
first type considered above, in v and x. An equation (aB <> bA)

(ax + by + c)dy/dx = Ax + By + C

may be reduced to this rule by first putting x + h, y + k for x and y,
and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.

(2) An equation in which y does not explicitly occur,

f(x, dy/dx) = 0,

may, theoretically, be reduced to the type dy/dx = F(x); similarly an
equation F(y, dy/dx) = 0.

(3) An equation

f(dy/dx, x, y) = 0,

which is an integral polynomial in dy/dx, may, theoretically, be
solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x,
y) corresponds, suppose, a solution [phi]1(x, y, c) = 0, where c is an
arbitrary constant; the product equation [phi]1(x, y, c)[phi]2(x, y,
c) ... = 0, consisting of as many factors as there were values of
dy/dx, is effectively as general as if we wrote [phi]1(x, y, c1)
[phi]2(x, y, c2) ... = 0; for, to evaluate the first form, we must
necessarily consider the factors separately, and nothing is then
gained by the multiple notation for the various arbitrary constants.
The equation [phi]1(x, y, c)[phi]2(x, y, c) ... = 0 is thus the
solution of the given differential equation.

In all these cases there is, except for cases of singular solutions,
one and only one arbitrary constant in the most general solution of
the differential equation; that this must necessarily be so we may
take as obvious, the differential equation being supposed to arise by
elimination of this constant from the equation expressing its solution
and the equation obtainable from this by differentiation in regard to
x.

A further type of differential equation of the first order, of the
form

dy/dx = A + By + Cy^2

in which A, B, C are functions of x, will be briefly considered below
under differential equations of the second order.

When we pass to ordinary differential equations of the second order,
that is, those expressing a relation between x, y, dy/dx and d^2y/dx^2,
the number of types for which the solution can be found by a known
procedure is very considerably reduced. Consider the general linear
equation

d^2y dy
--- + P-- + Qy = R,
dx^2 dx

where P, Q, R are functions of x only. There is no method always
effective; the main general result for such a linear equation is that
if any particular function of x, say y1, can be discovered, for which

d^2y1 dy1
---- + P--- + Qy1 = 0,
dx^2 dx

then the substitution y = y1[eta] in the original equation, with R on
the right side, reduces this to a linear equation of the first order
with the dependent variable d[eta]/dx. In fact, if y = y1[eta] we have

dy d[eta] dy1 d^2y d^2[eta] dy1 d[eta] d^2y1
-- = y1------ + [eta]--- and --- = y1------- + 2--- ------ + [eta]-----,
dx dx dx dx^2 dx^2 dx dx dx^2

and thus

d^2y dy d^2[eta] / dy1 \ d[eta] /d^2y1 dy1 \
--- + P -- + Qy = y1------- + ( 2--- + Py1) ------ + ( ----- + P--- + Qy1)[eta];
dx^2 dx dx^2 \ dx / dx \ dx^2 dx /

if then

d^2y1 dy1
---- + P --- + Qy1 = 0,
dx^2 dx

and z denote d[eta]/dx, the original differential equation becomes

dz / dy1 \
y1-- + ( 2--- + Py1)z = R.
dx \ dx /

From this equation z can be found by the rule given above for the
linear equation of the first order, and will involve one arbitrary
constant; thence y = y1 [eta] = y1 [int] zdx + Ay1, where A is another
arbitrary constant, will be the general solution of the original
equation, and, as was to be expected, involves two arbitrary
constants.

The case of most frequent occurrence is that in which the coefficients
P, Q are constants; we consider this case in some detail. If [t]*
be a root of the quadratic equation [t]^2 + [t]P + Q = 0, it
can be at once seen that a particular integral of the differential
equation with zero on the right side is y1 = e^[theta]x. Supposing
first the roots of the quadratic equation to be different, and [phi]
to be the other root, so that [p] + [t] = -P, the auxiliary
differential equation for z, referred to above, becomes dz/dx +
([t] - [p])z = Re^(-[t]^x), which leads to
ze^{([t]-[p])^x} = B + [int] Re^(-[p]^x)dx, where B is an
arbitrary constant, and hence to

(*) [t] = [theta]; [p] = [phi].
_ _ _
/ / /
y = Ae^([t]^x) + e^([t]^x)| Be^([p]-[t])^x dx + e^[t]^x | e^([p]-[t])^x | Re^-[p]^x dxdx,
_/ _/ _/

or say to y = Ae^[t]^x + Ce^[p]^x + U, where A, C are arbitrary
constants and U is a function of x, not present at all when R = 0. If
the quadratic equation [t]^2 + P[t] + Q = 0 has equal roots, so that
2[t] = -P, the auxiliary equation in z becomes dz/dx = Re^-[t]^x,
giving z = B + [int] Re^-[t]^x dx, where B is an arbitrary constant,
and hence
_ _
/ /
y = (A + Bx)e^[t]^x + e^[t]^x | | Re^-[t]^x dxdx,
_/ _/

or, say, y = (A + Bx)e^[t]^x + U, where A, B are arbitrary constants,
and U is a function of x not present at all when R = 0. The portion
Ae^[t]^x + Be^[p]^x or (A + Bx)e^[t]^x of the solution, which is known
as the _complementary function_, can clearly be written down at once
by inspection of the given differential equation. The remaining
portion U may, by taking the constants in the complementary function
properly, be replaced by any particular solution whatever of the
differential equation

d^2v dy
---- + P -- + Qy = R;
dx^2 dx

for if u be any particular solution, this has a form

u = A0 e^[t]^x + B0 e^[p]^x + U,

or a form

u = (A0 + B0x)e^[t]^x + U;

thus the general solution can be written

(A - A0)e^[t]^x + (B - B0)e^[p]^x + u,

or

{A - A0 + (B - B0)x}e^[t]^x + u,

where A - A0, B - B0, like A, B, are arbitrary constants.

A similar result holds for a linear differential equation of any
order, say

d^n y d^n-1 y
----- + P1 ------- + ... + P_n y = R,
dx_n dx^n-1

where P1, P2, ... Pn are constants, and R is a function of x. If we
form the algebraic equation [t]^n + P1[t]^n-1 + ... + P_n = 0, and all
the roots of this equation be different, say they are [t]1, [t]2, ...
[t]n, the general solution of the differential equation is

y = A1 e^[t]1^x + A2 e^[t]2^x + ... + A_n e^[t]_n^x + u,