Differential equation

“ Assuming the condition in its general form, take in the given differential equation a plane section of the surface φ = C parallel to the plane z, viz. put z constant, and consider the resulting differential equation in the two variables x, y, namely Xdx + Ydy = 0; let ψ (x, y, z) = constant, be its integral, the constant z entering, as a rule, in ψ because it enters in X and Y. Now differentiate the relation ψ (x, y, z) = ƒ (z) , where ƒ is a function to be determined, so obtaining
there exists a function σ of x, y, z such that
because ψ = constant, is the integral of Xdx + Ydy = 0 ”

“ 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. ”

“ The central discovery of the transformation theory of the solution of an equation F (x, y, z, dz/dx, dz/dy) = 0 is that its solution can always be reduced to the solution of partial equations which are linear. For this, however, we must regard dz/dx, dz/dy, during the process of integration, not as the differential coefficients of a function z in regard to x and y, but as variables independent of x, y, z, the too great indefiniteness that might thus appear to be introduced being provided for in another way. ”

“ Whether the forces are 'conservative' or not is immaterial. Since the differential equations of motion do not contain the time () in the finite form, if we eliminate from these equations, we obtain equations in and their differentials, the integration of which will introduce arbitrary constants which we shall call . If we can effect these integrations, the remaining constant () will then be introduced in the final integration, (viz., that of an equation containing ,) and will be added to or subtracted from in the integral equation. ”

“ The integral, therefore, is the appropriate mathematical expression for a theory of action between particles at a distance, whereas the differential equation is the appropriate expression for a theory of action exerted between contiguous parts of a medium.
We have seen that the result of the integration satisfies the differential equation. We have now to shew that it is the only solution of that equation fulfilling certain conditions. ”

“ The subject of singular solutions of differential equations has been materially advanced since the time of Boole by G. Darboux and Cayley. The papers prepared by these mathematicians point out a difficulty as yet unsurmounted: whereas a singular solution, from the point of view of the integrated equation, ought to be a phenomenon of universal, or at least of general occurrence, it is, on the other hand, a very special and exceptional phenomenon from the point of view of the differential equation. ”

“ 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 ”

“ The special relations which are contained in the above equations can all be summarized in a single differential equation, which is completely general for the function H of the three variables q, V, T. Namely, if we substitute in equation (46) for G the expression , and for R the value E + pV, it follows with respect to (10) the equation:
This differential equation represents the general expression for the application of the principle of relativity on the kinetic potential. Its general integral is expressed by (38) , of which one can easily convince himself. ”

“ If the original equation allow an infinitesimal transformation whose first extended form (see Groups) is Pƒ = ξdƒ/dx + ηdƒ/dy + η1dƒ/dy1, where η1δt is the increment of dy/dx when ξδt, ηδt are the increments of x, y, and is to be expressed in terms of x, y, y1, then each of Pω and Pω1 must be functions of ω and ω1, or the partial differential equation Πƒ must allow the group Pƒ. ”

“ The researches above referred to are closely connected with the theory of functions and of groups. Endeavours have thus been made to determine the nature of the function defined by a differential equation from the differential equation itself, and not from any analytical expression of the function, obtained first by solving the differential equation. Instead of studying the properties of the integrals of a differential equation for all the values of the variable, investigators at first contented themselves with the study of the properties in the vicinity of a given point. ”

“ A rational ordinary differential equation, with x as independent and y as dependent variable, is then one which equates to zero a rational function of y, the order k of the differential equation being that of the highest differential coefficient y (k) which enters; only such equations are here discussed. Such an equation P=0 is called irreducible when, firstly, being arranged as an integral polynomial in y (k) , this polynomial Irreducibility of a rational equation.is not the product of other polynomials in y (k) also of rational form ”

“ It may be true, as Herschel said, that in the twinkling of an eye a molecule solves a differential equation which if written out in full would belt the globe, but the molecule knows nothing of the equation—that is created by the mind, and as the modern discontinuous physics develops, it may be that we shall have to use difference equations rather than differential equations. But the differential equation expresses certain persistent relations between phenomena, and is thus real, and is the replica of an objective reality. ”

“ 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. ”

“ We rely, in fact, upon the theory of monogenic analytical functions (see Function) , a function being determined entirely by its development in the neighbourhood of one set of values of the independent variables, from which all its other values arise by continuation; it being of course understood that the coefficients in the differential equations are to be continued at the same time. But it is to be remarked that there is no ground for believing, if this method of continuation be utilized, that the function is single-valued ”

“ For groups of point-transformations in more than two variables there will be more than one set of differential invariants. For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise.
If a differential invariant of a continuous group of the plane be equated to zero, the resulting differential equation remains unaltered when the variables undergo any transformation of the group. ”

“ Or we can say, and it is just the same thing in other words:
(speed) equals the derivative of (distance) with respect to (time)
Now we can tell what a differential equation is. It is simply an equation in which a derivative occurs, such as the last example. Perhaps the commonest kind of equation in physical problems is the differential equation.
SOLVING PHYSICAL PROBLEMS
Now we were able to change the equation about floor area into other forms, if we wanted to find length or width instead of floor area. When we did this, we ran into the inverse or opposite of multiplication: division. ”

“ It is the only one that takes into account (and by that it is preferred over the other proposed extensions) those physical properties of space, that find their mathematical expression in the Maxwell-Hertz differential equation. ”