Equation

“ 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 pass on now to consider the solution of any partial differential equation of the first order; we attempt to explain certain ideas relatively to a single equation with any number of independent variables (in particular, an ordinary equation of the first order with one independent Partial differential equation of the first order.variable) by speaking of a single equation with two independent variables x, y, and one dependent variable z. It will be seen that we are naturally led to consider systems of such simultaneous equations, which we consider below. ”

“ 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. ”

“ 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. ”

“ In general, an indeterminate equation results when the number of unknowns exceeds by unity the number of equations. If, on the other hand, we have two equations connecting two unknowns, it is possible to solve the equations separately for one unknown, and then if we equate these values we obtain an equation in one unknown, which is soluble if its degree does not exceed the fourth. ”

“ 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. ”

“ We usually form much too vague an idea of what an equation is, when we give that name to every kind of relation of equality between any two functions of the magnitudes which we are considering. For, though every equation is evidently a relation of equality, it is far from being true that, reciprocally, every relation of equality is a veritable equation, of the kind of those to which, by their nature, the methods of analysis are applicable. ”

“ This equation enables us to conceive the circle under the general idea of a function of its radius, or of x, and consequently authorizes us to subject it to all the laws to which a function is subject, and leads us to the properties of their differentials, limits, and relations. By this equation we enter into infinitesimal calculus, the expressions of which present identity under a form which records a series of conceptions of long and profound analysis. ”

“ Difference-equations.—The summation of the series ... + un+2 + un-1 + un is a solution of the difference-equation Δvn = un+1, which may also be written (E − 1) vn = un+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—
vn+m + a1vn+m-1 + a2vn+m-2 + ... + amvn = N,
where a1, a2, ... am are constants, and N is a given function of n. ”

“ 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 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. ”

“ 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. ”

“ 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. ”

“ The integral of the general equation is, as we have seen, is an arbitrary function of the temperature , varying from one body to another; is a function of the temperature which is the same for all the substances of nature, and is a particular function of , and of satisfying the equation This function may be determined in the following manner. ”

“ The important principle was introduced by him that every equation results from the continual multiplication of as many simple ones as there are units in the index of its highest power, and has consequently as many roots as it has dimensions. ”

“ 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. ”

“ We may therefore speak of the plane and for many purposes can introduce the letter into our equations instead of For example, the equation is equivalent to the three equations It is to be noticed that on account of the indeterminateness of the and this method, regarded as an analytical artifice, is identical with that of Lagrange, also that in multiple algebra we should have an equation of precisely the same form as (3) to express the same relation between the planes, but that the equation would be explained to the student in a totally different manner. ”

“ 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. ”

“ 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. ”

“ Every Definition of a Line is an Equation. In order to complete the philosophical explanation of the fundamental conception which serves as the base of analytical geometry, I think that I should here indicate a new general consideration, which seems to me particularly well adapted for putting in the clearest point of view this necessary representation of lines by equations with two variables. ”

“ This transformation is called changing the independent variable from x to y.
Illustrative Example 1. Change the independent variable from x to y in the equation
Solution. Substituting from (35) , (36) , (37) ,
Reducing, we get
a much simpler equation. ”

“ A few problems lead to indeterminate equations of the third and fourth degrees, an easy indeterminate equation of the sixth degree being also found. The general type of problem is to find two, three or four numbers such that different expressions involving them in the first and second, and sometimes the third, degree are squares, cubes, partly squares and partly cubes, &c. ”

“ The fundamental equation of a surface which gives the value of the tension in terms of the temperature and potentials seems best adapted to the purposes of theoretical discussion, especially when the number of components is large or undetermined. ”

“ 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. ”

“ When in some particular cases the integrations of the preceding formula shall have been performed by substituting its expression in the equation (III) , the functions and will be determined by comparing them with those of the same order introduced by means of the different expressIons for ; so that this equation may become identical. All being thus determined, the densty given by the formula (2) will be known. ”

“ Thus we have obtained a value of the unknown quantity y, in terms of the known quantities q and r; therefore the equation is resolved.
3. But this is only one of three values which y may have. Let us, for the sake of brevity, put
A = −1/2 r + √ (1/27 q3 + 1/4 r2) , B = −1/2 r − √ (1/27 q3 + 1/4 r2) , Then, from what has been shown (§ 1) , it is evident that v and z have each these three values,
v = ∛A, v = α∛A, v = β∛A;
z = ∛B, z = α∛B, z = β∛B. To determine the corresponding values of v and z, we must consider that vz = −1/3 q = ∛ (AB) . ”

“ The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude, or at least an intermediate unit of which the value can be ascertained in terms of the ultimate standard. But though a ratio is the required result, an equation is the mode in which the ratio is determined and expressed. ”

“ 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. ”

“ We have then ⁠⁠ (6) By substitution of these values our fundamental equation becomes (7) where (the geocentric and heliocentric distances) are the only unknown quantities. From equations (6) we also get, by squaring both members in each,
(8) by which the values of may be derived from those of or vice versâ. ”

“ Here we have obviously to deal with a function which is such that its second differential coefficient is proportional to itself. The only function we know that has this property is the exponential function (see p. 143) , and we may be certain therefore that the solution of the equation will be of that form.
Proceeding as before, by multiplying through by , and integrating, we get , and, as
where is a constant, and . ”

“ 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. ”

“ Def.—If is the scalar quantity of something situated at a certain point the potential of for any point is a scalar function of defined by the equation
and the Newtonian of for any point is a vector function of defined by the equation Again, if is the vector representing the quantity and direction of something situated at the point the potential and the Laplacian of for any point are vector functions of defined by the equations 92. ”

“ Any one who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains. ”

“ When is defined as an implicit function of by means of an equation in the form
(A) , it was explained in the last section how it might be inconvenient to solve for in terms of ; that is, to find as an explicit function of so that the formulas we have deduced in this chapter may be applied directly. Such, for instance, would be the case for the equation
(B) . We then follow the rule:
Differentiate, regarding as a function of , and put the result equal to zero. [2] That is,
(C) . Let us apply this rule in finding from (B) .
; by (C) ; ; ; Ans. ”