a series of equations determining the successive differential coefficients of y, in the form the dependence of. f (x, y) upon f-1 (x, y), and hence ultimately upon f(x, y), being determined by the general equa tion Hence M and N being given, the expressions for are implicitly given also. dy dy Now &c. determine the coefficients of the several terms after the first in the development of y in ascending powers of x, by Taylor's theorem, or more generally in ascending powers of x-x, where x is a particular value of x. Leaving that first term arbitrary, the development is thus seen to be possible, and the result, while constituting the general integral of the given differential equation, shews that that integral involves an arbitrary constant. Actually to obtain the development, let (x) represent the general value of y, and let y, be the particular value of y corresponding to some particular and definite value, a, of the variable x. Then, writing (x) in the form But (x) is what y becomes when x=x. Hence (x) = Y。· dp (x) dy i.e. becomes when x=x. Again, '(x) is whạt dx Hence p'(x) = f(x, y) by (24). dx' In like manner "(x) is what dy dx2 becomes when x = x, and is therefore equal to f(x, y). Determining thus the successive coefficients of (29), we have finally If we assume x=0, and represent the corresponding value of y by c, we have y=c+f1(0,c)x+ƒ, (0, c) 1.2+ &c. ...... (31). Should however any of the coefficients in this development become infinite we must revert to the previous form, and give to x such a value as will render the coefficients finite, and therefore justify the application of Taylor's theorem. Virtually the integral (30) involves like (31) only one arbitrary constant. For in applying it we are supposed to give to x a definite value, and this being done the corresponding arbitrary value of y, constitutes the single arbitrary constant of the solution. [See the Supplementary Volume, Chapter XIX, Arts. 4 and 5.] ' (4) (1+y3) dx − {y+√/(1+y3)} (1+x2) a dy = 0. ✔ (5) sin x cos ydx — cos x sin ydy = 0. ✔ (6) sec3x tan ydx + sec2 y tan xdy = 0. ✔ 2. Different processes of solution present the primitive of a differential equation under the following different forms, viz. tan1 (x + y) + tan1 (x − y) = c, y2x2+1=2Cx. Are these results accordant? Yes, See a L.VT 3. Integrate the homogeneous equations: (1) (y-x) dy+ydx = 0. ކ (5) (8y+10x) dx + (5y +7x) dy = 4. Integrate the equations: (1) (2x-y+1) dx + (2y − x − 1) dy = 0. (2) (3y-7x+7) dx + (7y − 3x+3) dy = 0; v the former as an exact differential equation, the latter by reduction to a homogeneous form. → 5. Explain what is meant by variation of parameters, and, dy having integrated the equation x dx ay= 0, deduce by that method the solution of the equation x dy -ay=x+1. dx 6. Integrate, by the direct application of (23), the linear dy dx 7. Shew that the solution of the general linear equation +Py=Q may be expressed in the form 8. Shew that, (x) being any function of x, the solution of the linear equation dx represent dy by p, and then, differentiating and eliminating dx y, form a differential equation between pand x, that equation will also be linear. CHAPTER III. EXACT DIFFERENTIAL EQUATIONS OF THE FIRST DEGREE. 1. As the cases considered in the previous Chapter under which the equation Mdx+Ndy =0 is integrable by the separation of the variables, are but a small number of the cases in which a solution expressible in finite terms exists, Analysts have engaged in a more fundamental inquiry of which the following are the objects, viz. 1st, To ascertain under what conditions the equation Mdx+Ndy = 0 is derived by immediate differentiation from a primitive of the form f(x, y) = c, and how, when those conditions are satisfied, the primitive may be found. 2ndly, To ascertain whether, when those conditions are not satisfied, it is possible to discover a factor by which the equation Mdx+Ndy = 0 being multiplied, its first member will become an exact differential. These inquiries will form the subject of this and the following Chapter. PROP. I. The one necessary and sufficient condition under which the first member of the equation Mdx+Ndy=0 is an exact differential is Let it be considered in the first place what is meant by the supposition that Mdx + Ndy is an exact differential. It is that M and N are partial differential coefficients with respect |