written, not an algebraically new equation. Thus the process of reduction cannot be repeated. We have therefore y= Ce as the only common integral. [See the Supplementary Volume, Chapter XXII.] d'y 5. 3 dx3 dx2 + y = 0. + 4y = 0, it being given that one of the roots of the auxiliary equation, m3 — 3m3 + 4 = 0, is — 1. 6. dy-2x+2x-2y+y-1. = 8. What form does the solution of the above equation assume when k=1? is y = Ceasin ̄x, find the complete integral by the method of Art. 13. 13. The form of the general integral might in the above case be inferred from that of the particular one without employing the method of Art. 13. Prove this. 15. Explain on what grounds it is asserted that the complete integral of a differential equation of the nth order contains n arbitrary constants and no more. 16. Mention any circumstances under which it may be advantageous to form, from a proposed differential equation, one of a higher order. In deducing from the solution of the latter that of the former, what kind of limitation must be introduced? CHAPTER X. EQUATIONS OF AN ORDER HIGHER THAN THE FIRST, CONTINUED. 1. We have next to consider certain forms of non-linear equations. Of the following principle frequent use will be made, viz. When either of the primitive variables is wanting, the order of the equation may be depressed by assuming as a dependent variable the lowest differential coefficient which presents itself in the equation. Thus if the equation be of the form we have, on substitution, the differential equation of the first order, If, by the integration of this equation, z can be determined as a function of x involving an arbitrary constant c, {suppose z= p(x, c)}, we have from (2) If the lowest differential coefficient of y which presents itself be of the second order, the order of the equation can be depressed by 2, and so on. A similar reduction may be effected when x is wanting. Thus, if in the equation of the second order Should we succeed by the integration of this equation of the first order in determining p as a function of y and c, sup dy pose p=(y, c), the equation =p will give dx dy 2. In close connexion with the above proposition, stand the three following important cases. CASE I. When but one differential coefficient as well as but one of the primitive variables presents itself in the given equation. 1st. Let the equation be of the form by successive integrations day = √xdx + c dry doc = X, we have dx-1 We shall hereafter shew that the first term in the second member may be replaced by a series of n single integrals. generally integrable, but it is so in the case of n = 2. Thus Y, it is not there being given |