EXERCISES.

✓ 1. How are equations, in which all the differential coefficients have reference to only one of the variables, solved?

4. The partial differential equation of the first order which results from a primitive of the form u= f (v), where u and v are determinate functions of x, y, z, is necessarily linear. Prove this.

✓ 7. yp + xq = z.

E. x2p-xyq+y2 = 0..

9. Integrate the equation of conical surfaces

12. Required the equation of the surface which cuts at right angles all the spheres which pass through the origin of co-ordinates and have their centres in the axis of x.

It will be found that this leads to the partial differential equation of the last problem.

13. z-xp—yq=a (x2 + y2+z3)*. ✓

14. Find the equation of the surface which cuts at right angles the system of ellipsoids represented by the equation

Ax2+By+Cz2 = D3,

where D is the variable parameter. Lacroix, Tom. II. p. 678.

15. Find the equation of a surface which belongs at once Forsyth to surfaces of revolution defined by the equation py-qx=0, p. 3′51 Ex. 8 and to conical surfaces defined by the equation pa+qy=2.

In problems like the above we must regard the equations as simultaneous, determine p and q as functions of x, y, z, and substitute their values in the equation dz=pdx+qdy, which will become integrable by a single equation if the problem is a possible one, but not otherwise.

dz dz dz

16. х +y +t

dx dy dt

ху

=az +

t

Forsyth, p. 303, 4.6(1)

17. Explain the distinction between a complete primitive and a general primitive of a partial differential equation of the first order.

18. Find the complete and the general primitive of Forth,

292,2

2x.3.

22. Shew from the form of its integral that q=ƒ(p) belongs only to developable surfaces. ✔

23. Deduce two complete primitives of

pq=px+qy.

24. Deduce two complete primitives of

√p+√q= 2x.

of. Johnson, p. 324, Ex. 18 and 19.

25. Given two general primitives of a partial differential equation of the first order, in the forms,

shew that the dependence of the functions

(c) and 6 (a), when the two primitives lead to the same particular integral, may be determined by the following rule. Eliminate x and y from any four independent equations of the system

The two resulting equations will involve the relation required, and when the form of (a) is given, the elimination of a from both will give a differential equation for determining the form of y (c).

26. The equation z = pq has two general primitives,

d

1st. 2=

(y+a) {x+¤ (a)},

0 = ãa [{y + a} {x + $ (a)}],

d

2nd. 4z = {cx+2 + (c)}", 0 = √ {cx + 2 + ↓ (c)}* ;

dc

shew hence that the relation between (a) and (c) is expressed by the equations

[An interesting problem involving partial differential equations of the first order is discussed in the Supplementary Volume, Chapter XXXIII.]

CHAPTER XV.

PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

1. THE general form of a partial differential equation of the second order is

F(x, y, z, p, q, r, s, t) = 0 .........

(1),

It is only in particular cases that the equation admits of integration, and the most important is that in which the differential coefficients of the second order present themselves only in the first degree; the equation thus assuming the form

in which R, S, T and V are functions of x, y, z, p and q. This equation we propose to consider. The most usual method of solution, due to Monge, consists in a certain procedure for discovering either one or two first integrals of the form

u and v being determinate functions of x, y, z, p, q, and ƒ an arbitrary functional symbol. From these first integrals, singly or in combination, the second integral involving two arbitrary functions is obtained by a subsequent integration.

An important remark must here be made. Monge's method involves the assumption that the equation (2) admits of a first integral of the form (3). Now this is not always the case. There exist primitive equations, involving two arbitrary functions, from which by proceeding to a second differentiation both functions may be eliminated and an equation of the form (2) obtained, but from which it is impossible to eliminate

one function only so as to lead to an intermediate equation of the form (3). Especially this happens if the primitive involve an arbitrary function and its derived function together. Thus the primitive

2= $ (y + x) + + (y − x) − x {☀' (y + x) − f' (y − x)}..... (4), leads to the partial differential equation of the second order

but not through an intermediate equation of the form (3).

It is necessary therefore not only to explain Monge's method, but also to give some account of methods to be adopted when it fails.

[Part of the present Chapter is treated on a larger scale in the Supplementary Volume, Chapters XXVIII. and XXIX.]

2. It is not only not true that the equation (2) has necessarily a first integral of the form (3), but neither is the converse proposition true. We propose therefore, 1st, to inquire under what conditions an equation of the first order of the form (3) does lead to an equation of the second order of the form (2); 2ndly, to establish upon the results of this direct inquiry the inverse method of solution. And this procedure, though somewhat longer than that usually followed, is more simple, because exact and thorough.

PROP. 1. A partial differential equation of the first order of the form u = f(v) can only lead to a partial differential equation of the second order of the form

when u and v are so related as to satisfy identically the condition

For, differentiating the equation u = f(v) with respect to x,

dz

dp dq

and observing that x=P, dx

dx

=r,

dx

=s, we have