﻿IN 
  THE 
  DIFFERENTIAL 
  CALCULUS. 
  53 
  

  

  14 
  All 
  equations 
  of 
  the 
  form 
  y-mx' 
  —-^=X 
  can 
  be 
  converted 
  into 
  equations 
  

  

  1 
  d 
  x 
  n 
  

  

  . 
  d 
  r 
  v 
  

   m 
  *7' 
  

  

  

  For 
  

   Let 
  

  

  y-mx 
  ( 
  1) 
  .—^ 
  y 
  = 
  X 
  

   y= 
  /-D 
  - 
  v, 
  then 
  

  

  

  /Zd 
  . 
  v-mx 
  r 
  - 
  n 
  (—l) 
  n 
  l—D 
  + 
  nv=X. 
  

  

  or 
  

  

  or 
  

  

  

  or, 
  if 
  *'—« 
  = 
  *, 
  

  

  rf^ 
  

  

  wherefore 
  the 
  above 
  equation 
  is 
  reduced 
  to 
  an 
  ordinary 
  differential 
  equation 
  

   whatever 
  be 
  x, 
  provided 
  r 
  is 
  an 
  integer. 
  

  

  As 
  an 
  example, 
  let 
  us 
  take 
  the 
  simplest 
  case, 
  of 
  which 
  the 
  solution 
  will 
  be 
  

   found 
  at 
  pp. 
  257, 
  258 
  of 
  my 
  Memoir 
  on 
  General 
  Differentiation, 
  Part 
  III. 
  

  

  d 
  v 
  

   Ex.1. 
  y-a\/^lcc—± 
  = 
  

  

  Suppose 
  y=[—bv, 
  

  

  then 
  vjx 
  + 
  ajx 
  '- 
  ~_^t 
  (v 
  Jx) 
  = 
  CVx 
  

  

  z-ax 
  % 
  ^-=Cs/x, 
  where 
  s 
  = 
  v 
  jx 
  ; 
  

  

  or 
  z-ax 
  dx 
  

  

  2 
  

  

  z 
  — 
  e 
  

  

  2 
  

  

  I 
  a 
  aJ 
  x 
  J 
  

  

  e 
  kVx 
  f 
  A 
  C 
  /V.t 
  «V*\ 
  ,, 
  n 
  

  

  and 
  J/=/-D-^- 
  t~^~^V^ 
  e 
  )--( 
  L 
  ) 
  

  

  The 
  constants 
  A 
  and 
  C 
  are 
  not 
  both 
  of 
  them 
  arbitrary. 
  In 
  fact, 
  it 
  will 
  be 
  

   seen 
  that 
  0=i=, 
  so 
  that 
  the 
  final 
  form 
  of 
  equation 
  (1.) 
  is 
  

  

  