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  1. Path: sparky!uunet!portal!lll-winken!uwm.edu!zaphod.mps.ohio-state.edu!howland.reston.ans.net!spool.mu.edu!olivea!charnel!sifon!galois.math.mcgill.ca!labute
  2. From: labute@galois.math.mcgill.ca (John LABUTE)
  3. Newsgroups: sci.math
  4. Subject: Re: Can Anyone Solve this?????
  5. Message-ID: <1993Jan26.174333.17703@sifon.cc.mcgill.ca>
  6. Date: 26 Jan 93 17:43:33 GMT
  7. References: <1993Jan23.210732.19327@magnus.acs.ohio-state.edu> <1993Jan24.234937.12223@galois.mit.edu>
  8. Sender: news@sifon.cc.mcgill.ca
  9. Organization: McGill University
  10. Lines: 15
  11. Nntp-Posting-Host: galois.math.mcgill.ca
  12.  
  13.  
  14. The most elementary way to solve the differential equation y''-2y+y=0 is
  15. to introduce the differential operator D = d/dx. The equation then 
  16. becomes (D^2 - 2D + I)(y) = 0 where I is the identity operator. Factoring
  17. the polynomial in D the equation becomes (D - I)^2(y) = 0 which is
  18. equivalent to the equation D^2(e^{-x}y) = 0 from which follows
  19. e^{-x}y = a+bx and hence y = ae_x + bxe^x. These techniques only require
  20. linear algebra (algebra of linear operators) and elementary calculus and
  21. can easily be generalised to solve any constant coefficient n-th order
  22. linear differential equation.
  23.  
  24. -- 
  25. John Labute                                   Phone: (514) 393-3819
  26. Mathematics Department                          FAX: (514) 398-3899
  27. McGill University                            e-mail: labute@math.mcgill.ca
  28.