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Ex. 2. dy + 2y x dx = 2a x3 y3 dx.

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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.

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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) = c,

and the general form of which is

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(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

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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 2, 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) = c,

(79)

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

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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.

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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 C1 and C2; and to be of the form

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(85)

(86)

(df

= 0,

dz

(87)

df2

+ Q

dy

(d) + x (

df2

R

= 0;

dz

dx

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 f2, that is, of c1 and c; then multiplying the members of (87) severally by

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and therefore F(fi,f) satisfies (81): and therefore we have for

the general integral u = F(f1f2) = 0, ?

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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

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where (y) is the arbitrary function which enters into the general

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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

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or, what is equivalent, by means of the substitutions (80),

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either of which is the general integral and involves an arbitrary functional symbol.

It is useful to observe the geometrical interpretation of the process:

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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.

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