﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  45 
  

  

  

  9 
  v»- 
  ; 
  /_d_ 
  1 
  d_ 
  /u; 
  

  

  

  Hence 
  * 
  (-£) 
  = 
  -ij 
  

  

  2. 
  + 
  i 
  

  

  r 
  

  

  — 
  /X 
  + 
  l 
  

  

  /D 
  

  

  = 
  # 
  — 
  f== 
  — 
  a; 
  

  

  = 
  a? 
  

  

  Q— 
  l)r 
  / 
  — 
  r 
  ^ 
  -0»-l)i 
  

  

  

  Hence 
  *""(A 
  /V 
  «=(-!)* 
  * 
  r/ 
  (*" 
  +1 
  /-") 
  

  

  a 
  relation 
  between 
  differentials 
  with 
  positive 
  and 
  negative 
  values 
  of 
  r. 
  

  

  Section 
  II. 
  Application 
  of 
  the 
  preceding 
  Theory 
  to 
  the 
  Solution 
  of 
  

  

  Differential 
  Equations. 
  

  

  12. 
  The 
  first 
  example 
  which 
  I 
  propose 
  to 
  give 
  is 
  the 
  solution 
  of 
  equation 
  (B), 
  

   Art. 
  9. 
  

  

  Ex. 
  1. 
  -—^i 
  ( 
  x 
  * 
  -jt\ 
  -a 
  x 
  riJl 
  y, 
  where 
  fj. 
  may 
  be 
  anything 
  whatever. 
  

  

  Let 
  x 
  rfL 
  y 
  = 
  z, 
  

  

  d'- 
  1 
  

  

  -x 
  r 
  f 
  t 
  =-x- 
  r 
  rz 
  = 
  

  

  dx"- 
  1 
  d 
  * 
  

  

  z=a 
  z 
  

  

  • 
  _l 
  d^} 
  

   l 
  

  

  or 
  a"'- 
  — 
  , 
  ^- 
  1 
  Dr 
  r 
  '' 
  z=a 
  r 
  x 
  r 
  ~ 
  1 
  z 
  

  

  dx' 
  

   or 
  D(D-l) 
  .... 
  (D-r 
  + 
  2) 
  x 
  (D-r 
  fx 
  + 
  1) 
  e-' 
  z 
  = 
  a 
  r 
  e 
  rl> 
  e- 
  

  

  Let 
  - 
  or 
  e~ 
  6 
  z~f 
  ( 
  j 
  v 
  ; 
  then 
  

  

  D(D-l) 
  . 
  . 
  . 
  . 
  (D-r 
  + 
  2)x(D-rjtX4-l)/ 
  (- 
  -\v 
  = 
  

   a 
  r 
  e 
  ri 
  f 
  (~~)v 
  = 
  a 
  /(- 
  £ 
  + 
  1) 
  e 
  r 
  'v. 
  

  

  