Mathematical equation

“ By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable. Often in the consideration of physical relations one is able to build up an expression for the law governing the interaction of the parts or of the forces that govern them, such expression being naturally in the form of a differential equation, that is an equation containing differential coefficients with or without other algebraic quantities. ”

“ Thus, for example, in the case of rectifications, it will be first necessary to calculate dy, or dy and dz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only ds and dx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus. ”

“ And when such a differential equation has been found, one can get no further until it has been integrated. Generally it is much easier to state the appropriate differential equation than to solve it:–the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy. ”

“ When the linear differential equation, which we take to be of the second order, has variable coefficients, though there is no general rule for obtaining a solution in finite terms, there are some results which it is of advantage to have in mind. We have seen that if one solution of the equation obtained by putting the right side zero, say y1, be known, the equation can be solved. ”

“ We desire to form a rational function φ (η) , or say φ (A (y) ) , of η1, . . . η, in which the η2 constants Aij shall all be essential, and not reduce effectively to a fewer number, as they would, for instance, if the y1, . . . yn were connected by a linear equation with constant coefficients. Such a function is in fact given, if the solutions y1, . . . yn be developable in positive integral powers about x = a, by φ (η) = η1 + (x − a) n η2 + . . . + (x − a) (n−1) n ηn. Such a function, V, we call a variant. ”

“ In the most simple questions, and when we have really to resolve only a single equation with a single unknown quantity, this numerical resolution is none the less a very imperfect method, even when it is strictly sufficient. ”

“ Moreover, and this is the most difficult consideration of all, all these circumstances may be dependent upon the values supposed given to the arbitrary constants of the integral; in other words, the singular points may be either fixed, being determined by the differential equations themselves, or they may be movable with the variation of the arbitrary constants of integration. Such difficulties arise even in establishing the reversion of an elliptic integral, in solving the equation
(dx/ds) 2= (x − a1) (x − a2) (x − a3) (x − a4) ;
about an ordinary value the right side is developable ”

“ When λ1 is a positive integer, not zero, the second solution about x = 0 persists as a solution, in accordance with the order of arrangement of the roots of the index equation in our theory; the first solution is then replaced by an integral polynomial of degree -λ or −μ1, when λ or μ is one of the negative integers 0, −1, −2, ..., 1 − λ1, but otherwise contains a logarithm. ”

“ Thus, when we admit every species of functions into the definition of equations, we do not at all account for the extreme difficulty which we almost always experience in putting a problem into an equation, and which so often may be compared to the efforts required by the analytical elaboration of the equation when once obtained. In a word, the ordinary abstract and general idea of an equation does not at all correspond to the real meaning which geometers attach to that expression in the actual development of the science. ”

“ This equation, or equations, will be referred to as the principal equation, or principal equations, of the problem.
The problems and method of the calculus admit of more exact formulation as follows: We confine our attention to the case where the sought curve is plane, and the function F contains no differential coefficients of order higher than the first. Then the problem is to determine a curve joining two fixed points (x0, y0) and (x1, y1) so that the line integral
taken along the curve may be a maximum or a minimum. ”

“ It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. ”

“ This transformation can always be effected. On the other hand, every differential equation of any order in relation to a single function might evidently always be reduced to the first order, by introducing a suitable number of auxiliary differential equations, containing at the same time the different anterior derivatives regarded as new functions to be determined. ”

“ Let these n − 1 functions together with x1 be introduced as n new independent variables in all the r equations. Since the first equation is satisfied by n − 1 of the new independent variables, it will contain no differential coefficients in regard to them, and will reduce therefore simply to dƒ/dx1 = 0, expressing that any common solution of the r equations is a function only of the n − 1 remaining variables. Thereby the investigation of the common solutions is reduced to the same problem for r − 1 equations in n − 1 variables. ”

“ Putting each of the fractions in the characteristic equations equal to dt, the equations enable us, starting from an arbitrary element x′0, y′0, z′0, p′0, q′0, about which all the quantities F′, dF′/dp′, &c., occurring in the denominators, are developable, to define, from the differential equation F′=0 alone, a connectivity of ∞1 elements, which we call a characteristic chain ”

“ If then we were to subtract the smaller area from the larger, we should have left as a remainder the area , which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is the difference between the integral worked out for the superior limit and the integral worked out for the lower limit.
Let us then go ahead. First, find the general integral thus:
.
and, as is the equation to the curve (Fig. 52) ,
is the general integral which we must find. ”

“ The index equation at x = ξ1 is r (r − 1) + rψ (ξ1) / φ′ (ξ1) + θ (ξ) 1 / [φ′ (ξ1) ] 2 = 0, and if α1, β1 be its roots, we have α1 + β1 = 1 − ψ (ξ1) / φ′ (ξ1) and α1β1 = θ (ξ) 1 / [φ′ (ξ1) ] 2. Thus by an elementary theorem of algebra, the sum Σ (1 − αi − βi) / (x − ξi) , extended to the m finite singular points, is equal to ψ (x) / φ (x) , and the sum Σ (1 − αi − βi) is equal to the ratio of the coefficients of the highest powers of x in ψ (x) and φ (x) , and therefore equal to 1 + α + β, where α, β are the indices at x = ∞. ”

“ The artifice consists in rejecting ab initio all terms of the equation which belong to R. We do not form R at all, but only φ.Δx, or φ.dx, which is the differential dy. In the same way, in all applications of the integral calculus to geometry or mechanics we form the element of an integral in the same way as the element of area y.dx is formed. ”

“ Exceptions may arise when two solutions coalesce, or when infinite values of the variables occur.
Equations are of great importance in applied mathematics. The data of a problem generally lead to an equation, or a set of equations, among the quantities concerned. In practice a certain number of these quantities are known in any given case; the unknown quantities are then found by the solution of an equation or equations. Non-algebraic equations occur frequently—equations involving trigonometrical functions, for example. ”

“ In the majority of cases we only care for one of the roots of the equation, which is distinguished by being very small, and which will be most easily calculated by a small correction to the value which we get by neglecting the quadratic term. ”

“ When the number of unknown quantities exceeds the number of equations, the equations will admit of innumerable solutions, and are therefore said to be indeterminate. Thus if it be required to find two numbers such that their sum be 10, we have two unknown quantities x and y, and only one equation, viz. x + y = 10, which may evidently be satisfied by innumerable different values of x and y, if fractional solutions be admitted. ”

“ Slight reflection showed, that a problem elementarily solvable, must always possess the property of having the unknown lines in the figure relating to it connected with the known lines of the figure by an equation for the solution of which equations of the first and second degree alone are requisite, and which may be so disposed that the common measures of the known lines will appear only as integers. ”

“ An are arbitrary constants, and u is any particular solution whatever; but if there be one root θ1 repeated r times, the terms A1 eθ1 x + ... + Ar eθr x must be replaced by (A1 + A2x + ... + Arxr−1) eθ1 x where A1, ... An are arbitrary constants; the remaining terms in the complementary function will similarly need alteration of form if there be other repeated roots.
To complete the solution of the differential equation we need some method of determining a particular integral u ”

“ A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients. The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences. ”

“ The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x+Δx for x and y+Δy for y. If there is a differential coefficient Fundamental Artifice.of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx = 0 (R/Δx) = 0, as in § 7 above. ”

“ To integrate a system of simultaneous equations of the first order X1 = a1, ... Xr = ar in n independent variables x1, ... xn and one dependent variable z, we write p1 for dz/dx1, &c., System of equations of the first order. and attempt to find n + 1 − r further functions Z, Xr+1 ... Xn, such that the equations Z = a, Xi = ai, (i = 1, ... n) involve dz − p1dx1 − ... − pndxn = 0. By an argument already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation Xi = ai ”

“ In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. ”

“ From this equation and the n − 1 equations Πiω = 0, Pσω = 0, we can determine ω by a quadrature only. Hence can be deduced a much more general result, that if the group of r parameters be integrable, the complete system can be entirety solved by quadratures; it is only necessary to introduce the solution found by the first quadrature as an independent variable, whereby we obtain a complete system of n − r equations in n − 1 variables, subject to an integrable group of r − 1 parameters, and to continue this process. ”

“ Thus by our general theorem, if the differential equation allow a group of two parameters (and such a group is always integrable) , it can be solved by quadratures, our explanation sufficing, however, only provided the form Πƒ and the two infinitesimal transformations are not linearly connected. ”

“ It appears natural enough now to inquire whether, under proper conditions for the forms of the rational functions a1, ... an, it may be possible to ensure that in each of the series φ1, ψ1, [χ] 1, ... the number of negative powers shall be finite. Herein Regular equations. lies, in fact, the limitation which experience has shown to be justified by the completeness of the results obtained. Assuming n integrals in which in each of φ1, ψ1, χ1 ... the number of negative powers is finite, there is a definite homogeneous linear differential equation having these integrals ”

“ But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the ’s, but all such terms as ; and this will not be the same as , because is not a constant. For some of the ’s will be multiplied by big values of , and some will be multiplied by small values of , according to what happens to be. So we must bethink ourselves as to what we know about this process of integration being the reverse of differentiation. ”

“ When in a given set of differential equations the number of equations is greater than the number of dependent variables, the equations cannot be expected to have common solutions unless certain conditions of compatibility, obtainable by equating different forms of the same differential coefficients deducible from the equations, are satisfied. ”

“ To elucidate these general considerations, we apply them to examples by means of which we show that the different integrals, in the series of the same equation of partial differences, are transformed into one another, and may be expressed under a finite form by definite integrals, which also contain one or several arbitrary functions. In the single case, in which the integral in series contains only arbitrary constants, every term of the series by itself satisfies the given equation, so that the general integral is found expressed by the sum of an infinite number of particular integrals. ”