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 z the dependent variable; only instead of an arbitrary constant we must add an arbitrary function of x. The final solution is x+y+z = y2¢ (x). 554 13,38. It sometimes happens that equations not belonging to the above class are reducible to it by a transformation. with respect to y, and adding dy dz Restoring to w its value integrating with respect to x, dx' and adding an arbitrary function of y, we have Now (2) being arbitrary, [p(x) da is also arbitrary, and may be represented by 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 P, Q, and R being given functions of x, y, z, or constant. dz Usually the differential coefficients and are repre dz 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. Mém. Berlin 1778 3. Oeuvres Vol. 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 du = qdv. 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, *) E. g. let the diff. eq. be z = = px+qy; then cling & from this and the relation, 8.g. d2 = pdx+gdy Lind: zdx+xd2 = q( y out x dy), or p and of which the solution, Chap. XIII. Art. 5, assumes the .298 form u = α, v = b.... a and b being arbitrary constants. .(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. of. Johnson ✓ PROPOSITION. A primitive equation of the form u = (v), 4.271, p. 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. 294-5 ise. to a linear eq. ..(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. For solving the latter equation we have u= $ (v). It is also equivalent to F' {x, y, z, $ (v)} = 0, ø being an arbitrary, but Fa definite functional symbol. For solving the latter equation with respect to (v) we have a result of the form $ (v) = F(x, y, z), or $ (v) = u on representing F, (x, y, z) by u. Thus the proposition affirmed amounts to this, viz. that any equation between x, y, and which involves an arbitrary function will give rise to a linear partial differential equation of the first order. 2 Differentiating the primitive u = (v), first with respect to x, secondly with respect to y, we have The elimination of i A =9'B first, we have Eliminating '(v) by dividing the second equation by the If A, B, A', B'are lineon dy dz + 2 du du dx + dz P dx + dz. we hav = dx dz, du dv dx dy dy dx 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 Substituting these values in (6) there results, lp + mq = 1, which agrees with the result before obtained. 4. The general equation (6), of which the above theorem is a direct consequence, has been established by the direct elimination of the arbitrary function. But the same result may also be established in the following manner, which has the advantage of shewing the real nature of the dependence of the coefficients P, Q, R upon the given functions u and v. [See a Note at the end of the volume.] Differentiating the equation u = (v) with respect to all the variables, we have du = q(v) dv, or du du dx+ dx dy dy + du dz dz = &' (v) (de ex dx+ and as this equation is to hold true dv dv dy dy + dz dz)... (7), independently of the form see p.4 of the function (v), and therefore of the form of the derived function '(v), we must have du du du dx+ dy+ dz =0 dx dy dz ... du dv du dv ̄ du dv du dv du dv du dv (9). dy dz dz dy dz dx dx dz dx dy dy dx the dependent, Introducing now the condition that z is x and y the independent variables, we have pdx+qdy=dz. To eliminate the differentials, let the terms of this equation be divided by the respectively equal members of (9), and we have |