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Ex. 2. dy +2y x dx = 2a y3 dx.

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. f'(4) Oh + f() $(x) = f(x).


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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) = 0,

(75) and the general form of which is


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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

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- dx = dy
.. Qdx Pdy = 0;

(77) 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

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where P, Q, R are or may be functions of x, y and z, and in which 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) = 0,

(79) where y denotes the arbitrary function, which the complete integral requires ; then, by the process of Art. 50, Vol. I,

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substituting which in (78) we have

(81) and this is in the most general form of a partial differential equation of the first order and degree. It is the integral of this that we shall investigate.


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Now of (79) the differential is

la ) dx + (4) dy + (44) dz = 0; from which and (81) we have


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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 C, and cz; and to be of the form fi(x, y, z) = (1, f2(x,y,z) = (y

(85) where c, and c, are arbitrary constants.

Then from these we have

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on comparison of which with (81) it appears that either fı or fa satisfies (81): and so also will any arbitrary function of fi and fa: for let F represent an arbitrary function of fı and f, that is, of c, and cz; then multiplying the members of (87) severally by (de) and (dr ), and adding, we have

Plant) + 2 () +R () = 0; and therefore F($1,82) satisfies (81): and therefore we have for the general integral u = f(f1f2) = 0,2

= F(C1, C,) = 0;


or, as it may be expressed,
G1 = f (C2), 2

(89) fi = f(f); 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 (43) = f(x,y);

(90) then z = \f(x, y)dx++(y),

(91) where (y) is the arbitrary function which enters into the general integral, and which has y only for its subject. Similarly, if

(47) = f(x,y); 2 = \f(x, y)dy+(x).

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Ex. 1. a(z) +0 () = c; or, what is equivalent, by means of the substitutions (80),

ales) +0 ( + c (en) = 0. (92) Now by the conditions (81) we have


.:. bx - ay = (1, cx— az = (z;

(94) and therefore by reason of (89), bx -- ay = f (cx az);

(95) or u = f(bx ay, cx az) = 0);

(96) 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 cı and cy, the particular values of which determine the particular intersecting planes.

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