(2) y-xp=a (y2+ p), and y − xp = b (1 + x2p). 7. How many first, second, third, &c, integrals, belong to the general differential equation of lines of the second order given in Art. 11, and how many of each order are independent? 8. From the equation (y — b)2 = 4m (x − a) assumed as the primitive, deduce 1st the differential equations of the first order involving a and b as their respective arbitrary constants; 2dly the general functional expression for all differential equations of the first order derivable from the same primitive. 9. Of what primitive involving two arbitrary constants would the functional equation Þ(y — 2px, p3x) = c represent all possible differential equations of the first order? 10. How many independent differential equations of all orders are derivable from a given primitive involving x, y, and n arbitrary constants? CHAPTER II ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE BETWEEN TWO VARIABLES. 1. THE differential equations of which we shall treat in this Chapter may be represented under the general form M+Ndy. dx 0, M and N being functions of the variables x and y. In this mode of representation x is regarded as the independent variable and y as the dependent variable. We may, however, regard y as the independent and x as the dependent variable, on which supposition the form of the typical equation will be For as any primitive equation between x and y enables us theoretically to determine either y as a function of x, or x as a function of y, it is indifferent which of the two variables we suppose independent. It is usual to treat this equation under the form Mdx+Ndy=0, not however from any preference for the theory of infinitesimals, but for the sake of symmetry. The order of this Chapter will be the following. As the solution of the equation, if such exist, must be in the form of a relation connecting x and y, I shall first establish a preliminary proposition expressing the condition of mutual depend ence of functions of two variables; I shall then inquire what kind of relation between x and y is necessarily implied by the existence of a differential equation of the form I shall discuss certain cases in which the equation admits readily of finite solution; and I shall lastly deduce its general solution in a series. PROP. 1. Let V and v be explicit functions of the two variables x and y. Then, if V be expressible as a function of v, the condition dV dv dV dv dx dy dy dx = 0........ (1) will be identically satisfied. Conversely, if this condition be identically satisfied, V will be expressible as a function of v. 1st. For suppose V= (v). Then And this is satisfied identically; since by the process of elimination the second member vanishes independently both of the form of v as a function of x and y, and of the form of Vas a function of v. 2ndly. Also if the above condition be satisfied identically, V will be expressible as a function of v. For whatever functions V and v may be of x and y, it will be possible by elimi nating one of the variables x and y to express Vas a function of the other variable and of v. Suppose for instance the expression for V thus obtained to be But, v being by hypothesis an explicit function of both dv dy identically. Therefore p (x, v), which represents V, does not contain x in its expression; and V reduces simply to a function of v. Ꮳ We have supposed each of the functions and v to contain both the variables x and y. But, whether this be or be not the case, the identical satisfying of (1) is the necessary and sufficient condition of the functional dependence of V and v. For suppose either V or v, and for distinction we shall choose v, to be a function of one of the variables only, as x; and V to be a function of v. Then is also a function of x, dv dV and as and vanish identically the condition (1) is satisdy dy fied. Conversely, supposing v to be a function of x only, and (1) to be identically satisfied, that equation reduces to whence Vis expressible as a function of v. 2. The equation M+Ndy = 0 always involves the existence dx of a primitive relation between x and y of the form f(x, y) = c, in which c is an arbitrary constant. Let us first consider what is the immediate signification of the equation M+N = 0..... dx (1). We know that if Ax represent any finite increment of x, and Ay the corresponding finite increment of dy y, will represent dx approaches as Ax approaches Let us then first examine the interpretation of the equation We have M Ay The second member of this equation being a function of x and y, since M and N are functions of those variables, we may write the form of (x, y) being known when M and N are given. Now if we assign to x any series of values, it is possible to assign a corresponding series of values of y, any one of which being fixed arbitrarily all the others will be determined by (3). |