EQUATIONS, DIFFERENTIAL. 



EQUATIONS, DIFFERENTIAL. 



m 



tion to an easier form in the following way : Let Y be a function of x > 

 and let T" mean di : (is, Ate. Take the following eeta of quantities : 



-y 



V 



In + (x, y, y', y ", &C.) = 0, write for each of j-, y,y',&c., the quan- 

 tity opposite to it in the list above. If the equation thus produced 

 can be integrated into Y = I|<X, then by eliminating x from ar=Y', y= 

 XY* Y, the relation between x and y which satisfies $(x, y, /, &c.) = 

 is found. 



24. Various substitutions will sometimes aid in the solution of an 

 equation, as y = ux, y=u~ l , = it", y = , and so on : sometimes, when 

 a couple of expressions of the form as + by + c, a'.c + b'y + c', occur, it 

 will be convenient to introduce two new variables derived from 

 equating the preceding to AI< + Bit 1 and A'K + B'IT. We have given the 

 most effective general methods ; the number of transformations * 

 which have succeeded in particular cases is very large. 

 ^ 25. When the equation is of the first order, but of a higher degree, as 



P. y" -r FI y"- ' + . + i\y' + P O = 0, 



the theoretical mode of proceeding, occasionally convenient enough in 

 practice, is to find the 71 values of y', say y'=<t> l (x, y), y'=<t> t (x, y), 

 4c. If these can be separately integrated, giving, say <^ (x, y, c) = 0, 

 ^,(.1, y, c) = 0, &c., then the complete primitive of the equation is 



26. When, in the last, r,, p,_i, ftc., are all homogeneous functions 

 of x and y of the same degree, proceea as follows : Reduce the equa- 

 tion to the form 



y"- + A, y' >+... + A.= 0. 



in which A,, &o., are of the form <j> (y : x). Let y = t.r and find, in 

 terms of t, the several values of y' or t -Kef, from the equation. Now 

 ( + xt' = <t>t, is an equation in which the variables can be separated. 

 Solve each equation, and proceed as in the last. 



27. The only equation of differences of the first order which can 

 be generally treated, at present, is the form An, P,n* =<3i, where 

 p, and Q, are functions of x, or its corresponding form u,+ \ p f M = Q,. 

 Supposing a; to be an integer, which is what is usually required, let 

 2u, denote T m -|_i + ...-m_i, where m is an arbitrary integer 

 chosen to start from. Accordingly A5, n,. The solution of 

 +i P* = <Jiis 



28. Equations of differences in their most general form have solu- 

 tions which appear to resemble the singular solutions of differential 

 equations : but there are important points in which the resemblance 

 fails. (' Differential Calculus Library of Useful Knowledge,' p. 738.) 



29. Let y<*' stand for the th differential coefficient of y. The 

 following expression 



ia absolutely integrable, independently of all relation between y and 

 *,if 



P. r,' + P," r,'" + ..<. =0, 



in which accents refer to complete differentiation with respect to .r. 

 And the integral is 



One remarkable case is this : each term p m j/ (n) is absolutely inte- 

 grable when P. is a rational and integral function of a- of a degree 

 lower than the mth. Thus y', xy", x.-y 1 ', a^j iY , are all integrable 

 functions ; and xy*', x"y { * , are also integrable. 



30. The equation y(*>=.<t>x is not only completely integrable, but it 

 may easily have all its intermediate differential equations found. For 

 instance, suppose y = <fx. Multiply this successively by 1, .r, a?, x*, x*, 

 and, by the last section, we have five integrable equatidns, leading to 

 the five differential equations of the fourth order which belong to 

 y = <fx. Treat the first result in the same manner with 1, x, x 1 , x*, 

 the second with 1, x, x', the third with 1, x, and the fourth with 1, 

 and we have the 10 equations of the third order. Proceed in the 

 same way with the results, taking care never to let any multiplication 

 enter which raises the coefficient as high as the order of differentiation 

 which it accompanies, and then will appear successively 10 equations 

 of the second order, 6 of the first, and finally the original primitive. 



The student must not be discouraged by finding that he docs not succeed 

 In tolving com which require detached artiflcei in the same manner as the 

 writer of the elementary work before him. The art of solying equations is made 

 erlilent : tbe ait of constructing equations which can bo solved is behind the 

 curtain. There are few or no mathematical works, with many examples, to the 

 writer* of which the proverb may not be applied, that " those who hide know 

 where to find." 



31. The following theorems will sometimes be of use. Let p. stand 

 lorjz'fxdj:. Then the nth integral of o>.r, or (fd.c\'<t>x, is the 

 following scries divided by 1.2.3. . . (n 1) 



and (m + l)fx~r.dx=x~ + 'p. -P. + .+ , 



If the integrations in the last section be made by this last rule as they 

 arise, the results will be given in terms of P , p,, &c., and there will be 

 no difficulty about the constants, which in the ordinary mode may 

 appear to enter in too great numbers. 



32. There are very few cases of equations of higher orders than the 

 first which can bo integrated in general terms. The equation y"=^y 

 gives 



which is iutegrable. Again, y" + oy=it, q and K being functions of 

 y, gives 



x= f 



J 



where log. w =fydy 



^(2/kw"rfy) 



and, by a change of the independent variable, y" + ry' + qy' 3 = gives, if 

 p and <j be functions of x, 



~^- where log. w= -fpdx. 



33. The term linear equation is generally applied to one of (lie 

 form 



in which p ; &c. are functions of x only. If a solution be found whirh 

 has not distinct arbitrary constants, it is not the most complete 

 solution, but is called a particular * solution. It is a most important 

 property of the linear equation that if n distinct particular solution* 

 can be found, say y=T,, y=Y, J , &c., then the complete solution is 



y = C I Y 1 -K>,Y,-r + O.Y. 



where c p c 2 , &c., are any arbitrary constants. 



34. The most remarkable case is that in which the coefficients are 

 constants, as in y<"> + ,)/<"-'>+ . . . +a,y=0. If the equation 

 t" + Oit"-' + . . . + a, = have n unequal roote, a, 0, y, &c., the jj 

 solution is 



but if there should be, say m roots equal to a, the term which must 

 belong to a in the solution is of the form 



(A,, -r A t * + . . . + A.-,*"-') "* 

 A pair o imaginary roots, p. 

 the term 



v /I, contributes to the solution 



/" (K cos vx + L sin v.r) 



Letters introduced without specific mention mean arbitrary constants. 

 85. The most remarkable particular case of the preceding ia 



y"+a"y = y=K cos cuc + L sin ax 

 B cos (ax + A) 



No one differential equation is of so much importance as this. 



36. When the equation is linear in all but an independent term, aa 

 y<"' + p,y <"-'> + . . . . + p.)/=x 



x being a function of .r, its solution can be deduced from that of the 

 linear form, in which x = 0, by a method the importance of which will 

 justify some account of it in a separate article. [VARIATION- of PABA- 

 METERS.] We give the following result of it : y" + a'y = i gives 



a y=sin ax J x cos axdx cos ax J x sin axdx 

 + K cos ax-fL sin ax. 



87. The completed linear form (take the second order for an instance) 

 .V"t p y-*- p !2/ = x can be integrated ( 29) one step if both sides be 

 multiplied by the factor M, which satisfies the purely linear equation 



This is another differential equation of the same order, but the difficulty 

 is much reduced by it. The general solution of the last is not wanted ; 

 any particular solution will do. That is, if a particular solution of 

 every linear equation could be found, the general solution of every 

 completed linear equation could be deduced. And it will generally 

 happen that when one particular solution can be found, to make an 

 integrating factor, enough of them can be found to give the complete 

 solution of the original equation. 



88. The form $ (y, y', y " ) = 0, in which x is wanting, may be reduced 

 to an equation of the first order, by making y' j, when it becomes 



* The word particular is here opposed to general. Note this, because .tome 

 writers have ncd pnrticvlar lolution to signify what is now generally called 

 singular solution. 



