917 



EQUATIONS, DIFFERENTIAL. 



EQUATIONS, DIFFERENTIAL. 



sis 



ing x, y, ;/ (or dy : dx) and all the rest of the constants ; and according 

 as we eliminate one or another constant, we have one or another of as 

 many such equations as there were constants. These are called differ- 

 ential equations of $ = 0, and they are said to be of the first order. * 



2. The word order refers to the number of differentiations, the word 

 degree to the highest power of the highest differential coefficient which 

 enters. Thus y'^ + i/'^x is a differential equation of the second order 

 and the fourth degree. Let the accents always denote complete 

 differentiation with respect to x. For partial differentiation we shall 



dip dip 



save room by writing as follows : <p' f forjjr, <p' for -y, &c. 



3. If we eliminate two of the constants between <p= 0, iff = 0, <t>" = 0, 

 we have a differential equation of the second order, and so on ; there are 

 altogether 4 n (n 1) differential equations of the second order, if we 

 hare n constants. And generally, when there are n constants, there 

 are as many differential equations of the with degree as there are ways 

 of taking m out of n things. Thus there is only one equation of the 

 nth degree. 



4. If there be a system of p equations between p + I variables, so 

 that p of the variables are functions of the remaining one, and if we 

 differentiate each equation once, we can eliminate p constants, and form 

 p equations of the first order. With p second differentiations, we can 

 eliminate p other constants, and so on. And if there were p + q 

 variables, we should have similar systems with q independent va- 

 riables. 



5. Equations of differences are formed in a corresponding way by 

 taking differences instead of differential coefficients. Thus suppose 

 , means u, a function of x, and <J (x, u , , a) = 0. Change ./ into 

 x + Ax, usually into x + 1, and let u, then become H, + 1( or 

 M, -t-Au, . If we eliminate a between ^ (x, u,, a) = and <J> (x+1, 

 , + 1 ,a) =0, we have an equation of the form $(x,u,, ,+ ,) = 0,or 



' \fi(s,u,, Au,) = 0,as we please, and this is an equation of differences of 

 the first order. But exactly the same equation would be obtained, if 

 for a we had substituted any function of cos (2ir.r), or of cos (2 j-f- Ax). 

 On this point see INVARIABLE. 



6. When there are more than two variables, say three, it 1s possible 

 to form an equation in which an arbitrary function of a definite 

 function of two of them shall be eliminated. Thus, if <p { r, y, z, i^a 

 (jc, y) | 0, in which z is an implied function of jc and y and a (x, y) a 

 given function ; we differentiate with respect to x and y separately, 

 and produce three equations involving x,y,z,if,,if t , ia (.<,/) and 

 i^'o (x, y). From these three eliminate the two last, if and <fi', and we 

 get an equation involving only r, y, z, z" ,, z" ,, in definite forms. This 

 is called a partial differential equation. 



7. The analogies between the constants of a common differential 

 equation and the arbitrary functions of a partial one must not be 

 relied on as capable of being carried all lengths. It is not, for instance, 

 universally tme that two arbitrary functions may be completely 

 eliminated by differentiations of the second order. 



We now consider common differential equations of the first order. 



8. Let y = $ (jr, c), and y' Qi (x, c), and let elimination give y' = 

 X(-''i y ). the differential equation. There is another mode of arriving at 

 the same result. Let y = <f> (x, c) give c = # (x, y), or = f, + #', . i/. 

 The constant here disappears by mere differentiation, and x (x, y) must 

 be identical with +', : *' , . 



9. The difficulty of returning to the primitive from the equation 

 y 1 = x (x, y) consists in that of reducing it to the form +', + *', . y" = 0. 

 Generally speaking, common factors are made to disappear, and the 

 restoration of these factors is a problem of exactly the same difficulty 

 as the solution of the equation. 



10. The quantity P + Qy 1 , I* and Q being functions of x and ;/, is 

 integrable at once when] p' r * =<)',. When this condition, which is 

 usually called the criterion of integrability, is satisfied, the integral is 



in which y u constant in the first integration, and x in the second. 



Any arbitrary constant may be added. Either of the following 



forms are also integrals of ^ (.r, y) + x (r, y) . y 7 , if the criterion be 



:<cd. 



11. The equation y"=x ( x > y) can ba ve no other primitive with an 

 arbitrary constant, except y=<t> (x, c) from which it was derived. 

 But it may have another solution which has not an arbitrary constant, 

 or even more than one : these are called singular solutions. A singular 

 solution makes x an( l x'r infinite if y be substituted from it in 

 terms of x. It also makes *', and +', . infinite in the same manner. 

 All the solutions of either or both 4* = and +' = are singular 

 solutions; and all thejiingular solutions are thus found. But though 

 the most prominent and useful singular solutions are sure to be 

 obtained from x' = and x' = i these equations may not contain 

 all singular solution*, nor do they give nothing but singular solu- 



tions. But any solution of x' = which makes x.'" + X X finite is 

 a singular solution of the equation y' = x^ 



12. The singular solution may also be found from the primitive 

 y = <t> (x, c), as follows. Eliminate c between y = <f> (x, c) and = <t>' e (x, 

 c), the result is the singular solution. This rule and the last are 

 subject to exception, when the singular solution takes the form 

 x = constant or y = constant. 



13. The geometrical character of a singular solution is as follows. 

 Ity = <t> (.r, c) be the equation of a family of curves, that is, of one 

 curve for each value of c, the differential equation y' = x (*", y) also 

 belongs to every member of that family. The singular solution is the 

 equation of the curve which touches every one of the family of 

 curves. 



14. The equation y = y'x + <p (y') is historically remarkable as having 

 led to the theory of singular solutions : it is called Clairawfa form. 

 The complete primitive is y = <xr+^c, belonging to a family of straight 

 lines : the singular solution is found by eliminating c between thjs 

 and x + <f>'c = 0. 



15. Singular solutions are either intraneous, contained in the general 

 solution as particular cases ; or extraneous, not so contained. These 

 two species do not differ in geometrical character. A remarkable 

 theorem of Cauchy discriminates between them, as follows : 



Let y = p be a solution of y'=x(*> y)> an <l l et be anv small quan- 

 tity. If, then, x being constant, 



1 dy 



be finite, y=p is an extraneous solution ; if infinite, intraneous. 



16. It is not often that the factor which makes an equation inte- 

 grable can be recovered in any general form. As soon as the solution is 

 obtained, it can always be found ; but the only" use of it is to find the 

 solution. There is always an infinite number of such factors, any one 

 of which will do. When in P + <jy 7 , the criterion p', Q'J is not = ; 

 yet if (t~ t (f'i <)'i) be a function of x only, and not of y, and if 

 J <j-' ( p', <j', ) dx- v, then v (p + qy 7 ) is integrable. 



17. The equation y' + ry = Q, in which p and Q are functions of x 

 only, is always integrable. The factor of integration is /'*", and the 

 complete integral is 



The equation y 7 = Py + Qy" is reduced to the preceding form by 

 making^ = z ' j and y 7 = Py + Qy* by making z ~ y* * . 



18. When P and Q are homogeneous functions of x and y, that is, 

 when they are both of the form x" ip (y : x), with the same value of n, 

 the integrating factor is (PX + <Jy)~'. But generally the following 

 method is more convenient. Having 



assume y : x v, from whence 



t|i 



to which common integration can be applied. The equation is now 

 reduced to one in which the variables are said to be separated. 



19. It is hardly necessary to say that the variables can be imme- 

 diately separated in p + Qy 7 = 0, whenever P and Q are both of the form 

 <pxxjn/. 



20. It often happens that, when direct processes would require the 

 previous solution of an equation of indefinite form, indirect processes 

 will succeed in reducing the question of solution to one of elimination. 

 Thus x=f (y/) is solved when y' is eliminated between this and 



and y=/y is solved when y' is eliminated between it and 



where x' means 1 : y'. 



21. The equation y = x if>i/' -t- <fnj' can be reduced to elimination, thus : 

 Let y'=z, then 



rf.e 'z T _ *'z 

 ik z<pz z<fz 



which is of the form integrated in 17. Eliminate z between the 

 result of this and y x<f>z + tyz. 



22. The equation y=<p (x, y') can be reduced to a form of the first 

 degree and elimination. Let y' = z, then Z=F + QJ / , where P and Q are 

 iff, (x, z) and <f>', (x, z). If this equation can be integrated, we elimi- 

 nate z between the result and y = <t> (x, z). The form xif> (y, y') may 

 be treated in a similar manner. Or generally, let the equation be 

 <t> (x, y, y') = 0; that is, <f> (x, y, z) = 0, which gives by differentiation a 

 form P + <Jz-t-Hz'=0. If from this we eliminate y, by substitution 

 from the preceding, we have an equation of the first degree as to z 1 ; 

 from which, if integration be possible, elimination will determine the 

 relation between x and y. 



23. Any differential equation whatever may have a chance of reduc- 



