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The conditions that v should be the third differential of a function da-3u, are
lo di do
19 do ..
+ &c. = 0 ... (6).
dy In a similar manner are found the conditions that v should be a differential of any order : the numerical coefficients follow the law of those of the Binomial Theorem in the case of a negative index.
These remarkable formulæ were first discovered by Euler (Comm. Petrop. Vol. vin.) in his investigations concerning maxima and minima. A more direct demonstration is given by Condorcet, in his Calcul. Integral. Ex. (1) Let v= d'u = «d'y – yd' «.
- do Then - = d'y = 42
Therefore the first equation of condition becomes
Y2 – d’y = 0, and is therefore satisfied. In the same way the second condition is also satisfied, and we find
du = xdy – ydx + C. (2) Let
v = d’u = a* dʻy + (a + 2) « dy dx + (ay + 2x) da* + (awy + x®)d’x.
Both the conditions (1) and (2) are satisfied in this case, and we find
du = m dy + axyda + x? dx + C.
In this case the conditions (3) and (4) are both satisfied, so that v is the second differential of a function, which is found to be
U = a xy - y2 + C.
(+) To find the condition that
Rdx? + Sdx dy + T dya should admit of a first integral. If we assume S = S, + S, this may be put under the form
(Rdc + Sidy) d x + (S dx + Tdy) dy; and in order that it may admit of a first integral, we must
dy dy dx But from the indeterminateness of dx, and dy this involves the conditions dR da
dT ds, dy – do'
dx dy dR d’S, d” T d’S, dy dx dy
d x dx dy and therefore
d’R d’T d’s d’S, d’S
deja + dix* dx dy + dx dy dx dy' which is the required condition
The complication of the formulæ when the order of the differentials rises above the second renders their application almost impracticable, and as the subject is not one of any practical importance, it is unnecessary to adduce other examples.
INTEGRATION OF DIFFERENTIAL EQUATIONS,
Sect. 1. Linear Equations with constant coefficients.
These form the largest class of Differential Equations which are integrable by one method, and they are of great importance, as many of the equations which are met with in the application of the Calculus to physics are either in this shape or may be reduced to it. Let
9 + Any = X, (1) d 20"
'da be the general form of a linear differential equation with constant coefficients; Aj, AQ ... Abeing constants, and X being any function of x. On separating the symbols of operation from those of quantity this becomes 16.) +,)*+&C.... +.}y = X or so...y=X, (2) as we may write it for shortness. Now by the theorem given in Ex. 5 of Chap. xv. of the Differential Calculus, the complex operation 1 ) is equivalent to
Cano-a) (-... Com-on),
an, ag... On being the roots of the equation f(x) = 0 ...... (3).
Hence performing on both sides of (2) the inverse process of plan) or -a) (-as) ... Co-an), we have
The result of this transformation is different according to the nature of the roots of (3).
ist. Let all the roots be unequal; then by the theorem given in Ex. 6. Chap. xv. of the Differential Calculus, the equation (4) becomes
yang lama - )*x+ n. (co-es)" x + ...
+0. (..- .) X........ (5)
where N = -
(a, - a, (a, – 03) ... (a; – an)' and similarly for the other coefficients.
But by the theorem in Ex. 11 of the same Chapter, id
€ -4,4 X = 620 Sdx e-9,2 X. dx
d x) A similar transformation being made of the other terms, we find y = N, 691 sdx e-(1,8 X + N. 6°22 sdx e-A2.2 X + &c.
+ N,€"sdx e-On X......... (6). It is to be observed that each of the signs of integration would give rise to an arbitrary constant; and that this must be added in each of the terms when the integrations are effected. The value of y would then appear under the form y=N, EQ X (sdx e-9,2 X+C) + N, 602X (fdx e-A2+ X+C,) + &c.
+ N, 6°x2 (sdx e-Ant X + Cn) ...... (7). C, C ... Cn being the arbitrary constants.
The functions Ceax which arise in the integration are called complementary functions.
2nd. Let r of the roots of the equation (4) be equal to a. Then by the Theory of the decomposition of partial