do not, in general, suffice to enable us by the elimination of the latter, to form a partial differential equation of the second order free from arbitrary functions. We see then, 1st, that partial differential equations do not arise from the elimination of arbitrary functions only; 2ndly, that even as respects this mode of genesis, no general canons exist similar to those which govern the connexion of ordinary differential equations with their primitives. On both these grounds it will be proper, in considering special classes of equations, to examine their special origin and to seek therein the clue to their solution. Solution of partial differential equations. 2. Before proceeding to general theories of the solution of partial differential equations, it may be noticed that there are some equations of which the solution may be directly reduced to that of ordinary differential equations. This is the case when the partial differential coefficients have all been formed with respect to one only of the variables. We can then integrate as if this were in fact the only independent variable, provided that we finally introduce arbitrary functions of the other independent variables in the place of arbitrary constants. Ex. 1. Given x+y dz dx = 0. Multiplying by dx, integrating with respect to x, and adding an arbitrary function of y, we have It is permitted in the above, and in all similar cases, to complete the solution by adding an arbitrary function of y, because, with reference to the integration effected, y is constant; and it is necessary to add such a complementary function in order to obtain the most general solution, because an arbitrary function of one of the variables is more general than an arbitrary constant not involving that variable. Involving no differential coefficient with respect to x, it may be treated as a linear differential equation of the first order in which y is the independent, and the dependent variable; only instead of an arbitrary constant we must add an arbitrary function of x. The final solution is x, x+y+z=y2+(x). It sometimes happens that equations not belonging to the above class are reducible to it by a transformation, dw =w, then we have =x+y, whence integrating dy with respect to y, and adding an arbitrary function of x, 10 = x2y + y2 + p (x). dz 3 Restoring to w its value integrating with respect to x, dx and adding an arbitrary function of y, we have Now (a) being arbitrary, f(x) da is also arbitrary, and may be represented by x(x), whence [See the Supplementary Volume, Chapter XXIV. Art. 1.] Linear partial differential equations of the first order. 3. When there are but three variables, z dependent, x and y independent, the equations to be considered assume the form dz dz P, Q, and R being given functions of x, y, z, or constant. This form we shall first consider. dz dz Usually the differential coefficients and are repredx dy sented by p and q respectively. The equation thus becomes The mode of solution is due to Lagrange, and was first established by the following considerations. Since z is a function of x and y, we have dz = pdx+qdy. Hence eliminating p between the above and the given equation, we have Pdz - Rdx = q(Pdy — Qdx). Suppose in the first place that Pdz - Rdx is the exact differential of a function u, and Pdy - Qdx the exact differential of a function v, then we have Now the first member being an exact differential, the second must also be such. This requires that q should be a function of v, but does not limit the form of the function. Represent it by '(v), then we have du = '(v) dv, whence The functions u and v are determined by integrating the equations Pdz-Rdx=0, Pdy - Qdx = 0, and of which the solution, Chap. XIII. Art. 5, assumes the form u = α, a and b being arbitrary constants. v = b....................(4), Dismissing the particular hypothesis above employed, Lagrange then proves that if in any case we can obtain two integrals of the system (3) in the forms (4), then u = (v) will satisfy the partial differential equation, in perfect independence of the form of the function . We shall adopt a somewhat different course. We shall first establish a general Rule for the formation of a partial differential equation whose primitive is of the form u = (v), $ u and v being given functions of x, y, and z. Upon the solution of this direct problem we shall ground the solution of the inverse problem of ascending from the partial differential equation to its primitive. PROPOSITION. A primitive equation of the form u = $ (v), where u and v are given functions of x, y, z, gives rise to a partial differential equation of the form Pp + Qq = R where P, Q, R are functions of x, y, z. (5), Before demonstrating this proposition we stop to observe that the form u = 4 (v) is equivalent to the form f (u, v) = 0, f(u, v) denoting an arbitrary function of u and v. the latter equation we have u = $ (v). . It is also equivalent to F {x, y, z, & (v)} = 0, For solving being an arbitrary, but F a definite functional symbol. For solving the latter equation with respect to (v) we have a result of the form on representing F(x, y, z) by u. Thus the proposition affirmed amounts to this, viz. that any equation between x, y, and z which involves an arbitrary function will give rise to a linear partial differential equation of the first order. Differentiating the primitive u = (v), first with respect to x, secondly with respect to y, we have du du du du dy dz 1 = $' (v) (d dv (dy + dz2). Eliminating p' (v) by dividing the second equation by the first, we have Now this is a partial differential equation of the form (5). For u and v being given functions of x, y and z, the coefficients of p and q, as well as the second member, are known. The proposition is therefore proved. As an illustration, we have in Ex. 1, Art. 1, u=x—lz, v=y-mz, whence |