■{-^l 



4-24 Mr Bateuian, Sohdion of a system of differential equations 



where p is written for pix), and P^ for P (0), the initial value 

 ofP(0. 



Multiplying equations (1) by e~^\ and integrating from to 

 X with regard to t we obtain the system of equations 



xp-P^^- \,p \ 



^q -Qo = \p - Xo^- 



xr — Pfl = ^29' — '^■i'}' 

 xs — So = ^sr — X^s ^ 

 from which the values of ^, q, r may be obtained at once. 



If Qu = jRo = 'So = • • • = 0, i.e. if there is only one substance 

 present initially, we have 



^^'x'+Xi' ^~ {x + \)(x + \o)' ^~{x + Xi) (x + Xa) {x + Xs) ' 



and for the /itli product 



/ \_ XiXq ... A,t_i-ro 



''^^~ix~+x,){x + x.;)... (x + Xn) ^ ^• 



Putting this into partial fractions, we have 

 V {x) = — — — H -— - + 



where Ci = 



X + Xi X + X2 '" X + Xn' 



Xi X2 • • • ^»-i Pq 



(X2 ~ ^1) (X3 ~ \) • • • (^M — ^1) 



X1X2 ... X,j,_iPo |- (6). 



' ~ (Xi - X3) (X3 - X„) . . . {Xn - X,) 



- etc. 



To obtain the corresponding function N (t) we must solve the 

 integral equation 



v(x)=l e-*< N {t) dt. 

 Jo 



Now it has been shown by Lerch* that there is only one 

 continuous function N (t) which will yield a given function v (x) ; 

 hence if we can find a function which satisfies this condition it 

 will be the solution of our problem. It is clear, however, that 



1 r" 



+ X Jo 



x + 

 hence the above value of v (x) is obtained by taking 



N (t) = CiB-^^* + 026-"'*+ ... CnC-^-^ (7), 



where the constants have the values given by (6). 



* Acta Mathematica, 1903, p. 339. 



