Ex. 2. dy + 2y x dx = 2a x3 y3 dx. 3 The preceding is only a particular case of the following more general differential equation which is evidently capable of solution by the same process; viz. SECTION 5.-The integration of partial differential equations of the first order and degree. 384.] We must now consider differential expressions of another character; those, viz., wherein a relation is given between partial derived-functions and the variables: the general forms of these are (3) and (4) in Art. 364. I shall at present take the case wherein the partial derived-functions enter linearly, and where the coefficients are functions of the variables, including of course the case where they are constant. First let it be observed, that a partial differential expression which arises from an implicit function of two variables of the form u = F(x, y) = c, and the general form of which is (75) (76) where P and Q are functions of x and y, although involving partial derived-functions, is in fact a total differential expression; for differentiating (75), we have which is a total differential equation of the form (24). Now, from the explanation of partial differential equations which has been given in Article 366, it follows that the integral of a partial differential equation of the first order and degree requires the introduction of an arbitrary function; and although the integral may be particular, yet it is not general without it. Since then a total differential equation of the form (77) may by an inversion of the process followed above be changed into a partial differential equation, so does its general integral require an arbitrary function: the method of determining it will be explained in Section 6 of the present Chapter: thereby also we shall be led to a solution of total differential equations still more general than that of the preceding Sections. Let us now consider a partial differential expression of three variables x, y, z, and of the form and z, and in which where P, Q, R are or may be functions of x, y z has been considered a variable dependent on two independent variables y and x. To discuss it however in the most general form, let us suppose the original function to be of the form u = F(x, y, z) = c, (79) where F denotes the arbitrary function, which the complete integral requires; then, by the process of Art. 50, Vol. I, and this is in the most general form of a partial differential equa tion of the first order and degree. It is the integral of this that we shall investigate. PRICE, VOL. II. 32 either of which equations, it will be observed, involves the other by reason of (83); and let us suppose two independent integrals of these equations to be found, and to contain two arbitrary constants C1 and C2; and to be of the form : (85) (86) = 0, dz (87) on comparison of which with (81) it appears that either f1 or f2 satisfies (81) and so also will any arbitrary function off, and f2 for let F represent an arbitrary function of f, and f, that is, of c1 and c2; then multiplying the members of (87) severally by and (a), and adding, we have dF (a) and ( df2 and therefore F(ƒ1,f2) satisfies (81): and therefore we have for the general integral u = F(ƒvƒ2) = 0,2 = F(C1, C2) = 0; (88) and either (88) or (89) is the general integral, because each contains an arbitrary function in its most general form. 385.] The process requires further development and illustration but it will be better first to consider and solve some particular examples. Suppose the given equation to be where (y) is the arbitrary function which enters into the general and this is the complete integral. We may also thus prove (91): replacing (d) in (90) by its value from (80), we have dz dx or, what is equivalent, by means of the substitutions (80), (95) 0; (96) bx-ay = ƒ (cx — az) ; or u = F(bx—ay, cx—az) = either of which is the general integral and involves an arbitrary functional symbol. It is useful to observe the geometrical interpretation of the process: : Let (95) or (96) represent a surface: then from (92) it appears that the normal to the surface at every point of it is perpendicular to a straight line, whose direction-cosines are proportional to the quantities a, b, c but as these determine the direction and not the position of a line, we can only conclude that every normal is perpendicular to one of a series of parallel straight lines and the successive positions of these lines may vary according to any law; which law however is not given by the differential equation, but is required for the integral equation of the surface: in fact the insertion of it is absolutely necessary; for otherwise the equation cannot represent a surface; and the geometrical form of the law is the equation to the director curve along which the parallel straight line moves, and generates the surface; and this surface is cylindrical. This is also manifest from (88) and (94); (94) are the equations to two systems of parallel planes respectively parallel to the axes of z and y: and the intersection of two, viz., one of each system, will give the generating line of the surface; and the line of intersection will of course vary according to the functional relation between c1 and c2, the particular values of which determine the particular intersecting planes. 1 |