﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  

  

  By 
  the 
  usual 
  process, 
  this 
  equation 
  becomes 
  

   ~D(~D-l)u-q*e 
  2<> 
  u-i(i 
  + 
  l)u 
  = 
  Q 
  

  

  or 
  (D 
  + 
  ») 
  (D-»-l)w-j 
  2 
  e 
  2 
  '» 
  = 
  0. 
  

  

  Let 
  u=e~ 
  u 
  /(- 
  2) 
  v 
  > 
  then 
  

  

  (D 
  + 
  i)(D-i-l) 
  *-''/(- 
  J) 
  v 
  -<f 
  e 
  - 
  i6 
  f(-^ 
  + 
  Y)e 
  26 
  v- 
  

  

  or 
  D(D-2^-l)/(_^t,-j 
  ? 
  »/(-5+l)^'« 
  = 
  

  

  an 
  equation 
  which, 
  by 
  omitting 
  the 
  function 
  of 
  0, 
  is 
  reduced 
  to 
  

  

  D(D-l>-? 
  2 
  e 
  2 
  ^ 
  = 
  (1.) 
  

  

  by 
  the 
  condition 
  

  

  .. 
  D 
  _ 
  D-2*-l 
  , 
  / 
  D\ 
  .-,, 
  

  

  D 
  . 
  1 
  

   ~T 
  + 
  , 
  + 
  2 
  , 
  /_D\ 
  

   _ 
  __D 
  1 
  ' 
  V 
  2; 
  

  

  2 
  + 
  2 
  

  

  / 
  D 
  . 
  1 
  

  

  / 
  2 
  2 
  

  

  47 
  

  

  and 
  «=c 
  

  

  / 
  D 
  . 
  1 
  

  

  D 
  1 
  

  

  2 
  + 
  2 
  

  

  cz 
  

  

  = 
  e 
  *• 
  ' 
  ' 
  — 
  , 
  — 
  e 
  v 
  

  

  /~t 
  

   =(- 
  2 
  r 
  * 
  i+1 
  (- 
  4-Y 
  - 
  

  

  \x 
  dx) 
  x 
  

  

  But 
  the 
  solution 
  of 
  Equation 
  (1.) 
  is 
  

   Hence 
  the 
  complete 
  solution 
  of 
  the 
  given 
  equation 
  is 
  

  

  M= 
  ^ 
  2) 
  * 
  \xTx) 
  x^ 
  6 
  +Be 
  ) 
  

  

  This 
  solution 
  is 
  susceptible 
  of 
  another 
  form 
  by 
  Art. 
  11; 
  for 
  by 
  that 
  Article, 
  

   nd\i 
  v 
  _, 
  lV 
  - 
  -(2<+2)/ 
  3 
  ay 
  t> 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  N 
  

  

  