

EQUATIONS, DIFFERENTIAL. 



EQUATIONS, DIFFERENTIAL. 



822 



<j> (y, z,:d; : dy). If z or y' be thence found in terms of y, the variables 

 can be immediately separated : if y be found in terms of z, we eliminate 

 z between y = <fa aa found and xJ (ty'zdz : z). 



39. As an d priori mode of constructing linear equations and their 

 solutions, note the following: If <f> (*, y, y 7 , . .. . y 1 " 1 ) have for its 

 solution y if* (x, a, b, r, . . .),a,t>, c, &c., being arbitrary constants, and 

 if ?, P,, P,, Ac., be the differential coefficients of <f> with respect to 

 !' y'> y"> ^ c -> considered as independent variables (with <fi substituted 

 in them instead of y : then the solution of the linear equation 



A, B, Ac., being new arbitrary constants. 



40. If to aa equation of the nth order, all the n equations of the 

 (n l)th order can be found, each with an arbitrary constant, there is 

 no occasion for any further integration. For if y', y", . . . . y ( * l) be 

 eliminated between the n equations, the result wUl be the complete 

 primitive. 



41. The purely linear equation, or indeed any one which is homo- 

 geneous with respect to y, y 1 , y", Ac., can be lowered one order by 

 assuming log. y=J~zdx, or y = /* (we have used the former, as more 

 convenient to print). If this be done with p,^" + p^y' + p s y = 0, it gives 



p,/ + VjP + p,z + p,= 0. 



42. In this way the solution of y" = $x . y is connected with that of 

 :' + :' = <t>.r. When <^- = ar", it is called Riccati's equation, and it can 

 be integrated in finite algebraical terms whenever = 4i : (2i 1), 

 t being a positive integer. 



48. If only a particular solution of y" + Py / -Kjy = can be found, 

 then the general solution of y " + ry' + <jy + R = can be found from it. 

 Let y= r be the particular solution of the first, and let log. w=/ prfx : 

 the general rotation of the second is 



-,//_ 



44. If f (x, y, /, y*) = would become homogeneous with respect 

 to x and y, by considering y, y 1 and y* aa of the dimensions , n 1, 

 2, assume y = /:, y > j^'v,y* = jc*~^v!, which reduces the equation 

 to the form ^ (u, *,) = 0. Then deduce 



(a it 1 r) <fn-(r H) dr. 



and substitute for if it* value in terms of u and r. If this equation of 

 the first order can then be integrated, giving, say x ("> *) 0, we have 

 to complete the process by integrating x(H x ~*> y V~ ( *~*>) = 0. 



45. To destroy the second term of y" + Py' + Qy + R=.0 proceed as 

 follows. Let l=fr-f**dx, and find x in terms of (. Then we have 



46. Any number of simultaneous equations which leave only one 

 independent variable can be reduced to .equations between two 

 variables. For instance, let there be three equations between x, y, z, (, 

 and differential coefficients of x, y, z, with respect to t. Now, looking 

 upon x, jr 1 , Ac. y, y', 4c. (excluding z, z 1 , Ac.), as BO many independent 

 quantities, differentiate the three equations with respect to t. There 

 are then: ntroduced two new differential coefficients of .<; and y, and 

 thret new equations : or the number of equations gains one upon the 

 number of quantities. Repeat the process, and the same thing happens 

 again, and so on. Consequently a step must arrive at which the 

 number of equations is made one more than the number of quantities. 

 All the quantities can then be eliminated, and there results one equa- 

 tion between z, t 1 Ac., and t. The same thing may be done with x and 

 y. The process will also reduce the number of equations to as many 

 M there are independent variables, whatever that number may be. The 

 order of the result is the sum of the orders of the original equations. 



47. In the simple case of A/ + Bz / + c = 0, :,ry' + & + B.=0, when 

 A, f, Ac., are functions of x, y, z, and if is dz : dx, Ac., we may pro- 

 ceed by eliminating y' and forming say z'=M, a function of :r, y, z. 

 Hence z"=M'. + M',y' -m'.z', from which we may eliminate y and 

 y' l'_v the original equations, and thus produce an equation of the 

 second order to determine z in terms of x. And the same fur y in 

 terms of x. 



43. Let ( be the independent variable, and let . 



uns ti 'in of ( only. Determine and z from 



t will generally be a function of double value ; let , and , be its two 

 values, and z, and z ( the corresponding tv,-o values of z. Finally, 

 determine x and y from 



49. When the forms are 



A, &c. being constants (a case which is often useful) determine the 

 two values of 9 from 



and the two values of z from 



, + 0,9 



and find x and y from x + 0, y = 2, , x + 6 t y=z t . 



50. Also, when the simultaneous equations are linear, with constant 

 coefficients, proceed as follows. Say there are four variables, x, y, z, t ; 

 and three equations. Assume xt*', y = i"', z Cf"', and substitute : 

 the result will be, after division by t*', three equations of the form 

 ^ (a, 6, c) = 0, in which 5 and c only enter in the first degree. Hence 

 the values of o can be found by an equation whose degree is the sum 

 of the orders of the original equations, with one value of b and c for 

 each value of a. Let the values be o,, a v b lt b v e v c v Ac., then the 

 complete solution is 



c, Ac. being arbitrary constants. 



61. Whenever the solution of a differential equation depends upon 

 that of an ordinary equation, and supposing the roots to be all diffe- 

 rent, the form of the solution is 



the solution, though still a solution, ceases to be the complete solution 

 when there are any sets of equal roots. If there be m roots equal to a, 

 and if <p x, <t>" x, Ac., be the successive differential coefficients of <jxr, 

 this set of roots contributes to the value of x the terms 



of which a particular case has been seen in 34. 



52. We now look at one equation with three variables, two of wliich 

 are therefore independent. Let xdx + ldy -f zrfz=0 be the equation. 

 This may have arisen direct from ip(x, y, z) = f, without alteration or 

 loss of factors, Ac.: and this will be the case when x', =T', T', = z' y , 

 z'. =x',, and then only. To return to the primitive, integrate 

 xdx + idy ( 10) as if z were constant, and let the result be P. The 

 integral of xrfx + vrfy + zrfj is then 



Or thus : when the criteria are satisfied, the integral (omitting the 

 usual commas, to save room) of 



<p (xyf) dx + X (xyz) dy + $ (j-yz) rfz is 



in the first of which y and z are treated as constants, and z in the 

 second. 



58. If the criteria be not satisfied, a factor M may possibly exist, 

 such that MX rf.< + Hi dy + MZ dz shall be integrable. But this cannot 

 happen unless the following criterion be satisfied : 



and the factor, when this is satisfied, must satisfy any two of the 

 three equations. 



M (x\ Y',) + XM\ TM', =0 



M (Y', Z',) + YM', ZM', =0 



M (z', x',) + ZM', xnV=0 



But if the criterion be not satisfied, then there is no equation what- 

 soever between x, y, z, which always produces xcfa+Ycy + zifc = 0. 



54. The equation ndx + Ydy + zdz 0, when produced from 

 <f> (.r, y, z) = c, is the equation of a family of surfaces, the individuals of 

 which are defined by the different values of c. That is to say, any 

 point of one of these surfaces being taken, and any other point 

 infinitely near to it, the equation is satisfied by x, d.r, Ac., as derived 

 from these points. But when the criterion is not satisfied, then 

 Jidx + tdy + zdz still belongs to any surface in the following limited 

 sense. On any surface, and through any point of it, a curve may be 

 drawn such that the equation x</^ + Ac. is satisfied if the two con- 

 tiguous points first named be taken on that curve. 



55. When there are three variables and two equations, with two 

 constants, as in $ (x, y, z, a, 6)= 0, <J- (x, y, z, a, b,)= ; the most ready 

 theoretical mode of imagining the differential equations formed is the 

 reduction to the form o=* (x, y, z),]b = V (x, y, z), and differentiation. 

 Two equation* of the form i'dx + <)dy + Kdz=0 are thus produced; 

 from which, by combination, may be derived an infinite number of 

 pairs of the same kind, answering to the infinite variety of pairs which 

 can be produced from the primitive equations. This corresponds 

 with the obvious geometrical fact that one curve may be the inter- 

 section of an infinite number of different pairs of surfaces. 



