Differential Equations and Their Importance to Students of Physics and Chemistry FRED W. SCHUELER University of Colorado, Boulder, Colorado
"F
IRST impressions are the most lasting." This statement must claim as one of its greatest. exceptions the subject of mathematics. One need only recall the numerous mathematical paradoxes dealing with the infinite to appreciate how false first impressions based upon pure intuition may be. Consider as a further example the dilemma of the college junior who is casting about for a course that will logically follow his first course in calculus, and who finds his attention called to the subject of differential equations. If he is a bright student and inquires as to the nature of this subject, he is often rewarded by the presentation of afew typical differential equations, e. g.,
dynK
(a)
dx
$ + ra dydr- +
(COS
~ ) = y tan z
(')
in which he is told that the problem of differential equations consists in discovering the function y = f(x) such that (in a) the first derivative of y is constant. He may quickly recognize this as the linear form y = kx b, and the latter seems a t least as general as its corresponding differential equation. , Upon examining the example ( b ) he is not impressed, for his first idea may be that any function of which second derivative plus xZ times its first derivative plus cos x times y
+
comes out identically equal to tan x, must be of a very special sort. Indeed, if the student's interest is in the application of differential equations to chemistry, he may seek the solutions of such special equations as the following: Laguerre's (1834-1866) : d'w de'
dw -+nw=O dz
s-+(I-e)
Legendre's (1752-1833) :
Mathieus' (1835-1890) :
$ + + 169 (a
cos 22)
w
=
0
Bessel's (1754-1846) : ~ *d'w~ + z dw - + ( z ~ - n ~ ) w = O dz dz
Gauss' (1777-1855): d2w
dl - 2) p
+
[Y
- (a + P
+ lbl dw - -a@w dz
Hermite's (1822-1905): d'u
dez
dw +2nw dz
2 2
= 0
=
0
Though these equations are of extreme importance in theoretical chemistry (1, 2 ) , the student may be disappointed by their apparent special nature, and his feeling is not alleviated upon being told that all are special cases of the general Lam6 (1795-1870) equation.
and, as further illustrations consider the following: 1. When a = 1, j3 = 1, y = 2, z = -z and the whole series is multiplied through by e, one obtains
2.
z
W h e n a = 1, ,9 = 8 , y = 1 , z = - ( B +
B
- 1 3
the hypergeometric series reduces to That superficial appearance has led him far astray from the truth becomes apparent to the student who discovers that while the general Lam6 equation is still but a special case of the general second order differential equation,
In fact, the hypergeometric series may, under various conditions imposed upon the constants a , 0, y, and the variable z, represent nearly all of the functions of elementary mathematical analysis, e. g., sin z, cos z, log (1 z), d,etc. (3). The above discussion, therefore, indicates that in spite of the impression of extreme specialty that one gets upon his first contact with the subject of diierential equations, this impression, as are many Grst impressions of a mathematical subject, is completely false. That is to say, the comprehensive study of a single diierential equation (e. g., the Gauss hypergeometric equation) yields a knowledge of functions including, and far in excess of, all the functions described in elementary algebra and first courses in the calculus. I t is easy to understand, therefore, why many European universities devote as much as a full half year to some such "special" diierential equation as the Gauss' hypergeometric'equation. It is the belief of this writer that a single course in diierential equations, which emphasizes the generality of the subject, will do more to train chemistry and physics majors in the use of mathematical functions pertinent to their field than any other single mathematical course after the calculus.
+
yet, it represents an order of generality far exceeding anything which he has studied in all of his previous mathematics. To illustrate just how general the Lam6 equation is, consider one of its special cases, i. e., the Gauss hypergeometric equation given above. The Gauss hypergeometric equation has for its solution the function W = a F ( q p, y; z) given in the form of the infinite series,
which converges, in general, for lei < 1. To see just how general this special case of the general Lame equation is, choose the constants a = 1, and B = 7; then one gets,
which is the ordinary geometric series; hence the name hypergeometric series for the more general series and hypergeometric equation for its associated differential equation. Further, if one chooses the constants a = -n, p = 8, y = P, z = -2, then the hypergeometric series reduces to the simple binomial expansion or binomial theorem,
REFERENCES
(1) GLASSTONE, SAMUEL,"Theoretical Chemistry," D. Van Nostrand Co., Inc., New York, 1944. (2) EYRIFG,H., J. WALTER,AND C. E. KIMBALL,"Quantum Chemistry," John Wiley and Sons, Inc., New York, 1944. (3) WI?TAKER,E. T., "Modern Analysis," Cambridge Univers~ty Press, Cambridge, Mass., 1902, pp. 240-65.
As we gained mastery over the forces of nature, rue took part of the dividends in better living and part in more leisure or less employment. As technology adoances, i t is reasonable to suppose that these trends will continue. Let us be done with this scatterbrain doctrine that the end of alland be all of life is work, toil, or employment.John W . Scoville in National Business and Financial Weekly
