(y2 + yz + z2) dx + (x2 + xz + z2) dy + (x2 + xy + y2) dz Assuming x = uz, y = vz, we have on reduction whence integrating (v + 1) du+ (u + 1) dv uv + u + v u + v + 1 log z = log + C. uv + u + v Finally we have xy +xz+yz = C', x + y + z for the complete primitive. The last two equations might have been integrated without preliminary transformation. (Lacroix, Tom. II. pp. 507–510.) Integrating factors. 7. The equation Pdx+ Qdy + Rdz=0 can also, when there exists a single complete primitive, be integrated by means of a factor. If u be that factor, then, since the expression μ μPdx+pQdy + μRdz' Multiplying these equations by P, Q, and R, respectively, adding, and dividing by μ, we have the same equation of condition which was before obtained. When this equation is satisfied a particular form of the factor μ will frequently suggest itself. In Ex. 3 the functions (ay-bz)*' (cz - ax)2' (bx — cy)* are integrating factors. In Ex. 4 the functions. 1 (x + y + z)2 and 1 are integrating factors. Equations not derivable from a single primitive. 8. To solve the equation Pdx + Qdy + Rdz = 0, when the equation of condition (8) is not satisfied. In this case the solution consists of two simultaneous equations between x, y, z, one of which is perfectly arbitrary in form. For representing an assumed arbitrary equation in the form ƒ (x, y, z) = 0 ......................... and differentiating, we have (20), Now these two equations enabling us, when the form of f(x, y, z) is specified, to eliminate one of the variables and its differential, e.g. z and dz, from the equation given, permit us to reduce it to the form Mdx+Ndy = 0, M and N being functions of x and y. Solving this, we obtain an equation involving an arbitrary constant, and this equation together with (20) will constitute a solution. By giving different forms to f(x, y, z) every possible solution may be obtained. What a solution thus found represents in geometrical construction is the drawing, on a particular surface, of a family of lines, each of which satisfies at every point the condition Pdx+Qdy + Rdz=0. Now dx, dy, dz are proportional to the directing cosines of the tangent line. Hence the geometrical problem may be represented as that of drawing on a given surface a family of lines, in each of which the directing cosines cos p, cosy, cos x at any point shall satisfy the condition Pcos + cos + R cos x = 0 .(21). Ex. Required the most general solution of the equation which is consistent with the assumption that it shall represent a series of lines traced upon the ellipsoid whose equation is It will be found that (a) does not satisfy the equation of condition (8). indicating that the projections of the proposed family of lines will be a certain series of central conic sections. If a = b = c = 1 the proposed equation admits of a single primitive, viz. x2+ y2+z2=1. And any line traced on the surface of which this is the equation will satisfy the differential equation; for the equation (c) by which the lines are ordinarily determined is now reduced to an identity. The above method of solution is due to Newton. Monge has however remarked that the general solution may be expressed by the equations (10) and (11) of Art. 4, viz. by the simultaneous system where μ is the integrating factor, and V the corresponding integral of the expression Pdx + Qdy. It is indeed shewn in that Article that (22) does satisfy the differential equation provided that the condition (23) is satisfied. But there is no practical advantage in the employment of Monge's form. Applied to the problem of drawing on a given surface lines satisfying the condition expressed by the differential equation, it makes the determination of the arbitrary function (2) itself dependent on the solution of a differential equation. Thus in the example last considered we have, on giving to μ the value 2, To apply this to the problem of drawing lines satisfying the conditions of the problem on the ellipsoid it is necessary from the above three equations to eliminate x and y. From the second and third which here suffice, we have The particular solution sought is therefore expressed by the equations (e) and (ƒ), which are together equivalent to the previous solution expressed by (b) and (d). Total differential equations containing more than three variables. > 9. It will suffice to make a few observations on the equation with four variables Pdx+ Qdy + Rdz + Tdt = 0........................ (24), ........... and to direct attention to the general analogy. |