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by: Sydney Ezelle

ch1_math_238.pdf MATH 238

Sydney Ezelle

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ch1 notes & test review
Appld Diff Equations I
Tania Hazra
Study Guide
50 ?




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This 8 page Study Guide was uploaded by Sydney Ezelle on Wednesday September 14, 2016. The Study Guide belongs to MATH 238 at University of Alabama - Tuscaloosa taught by Tania Hazra in Fall 2016. Since its upload, it has received 3 views. For similar materials see Appld Diff Equations I in Math at University of Alabama - Tuscaloosa.


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Date Created: 09/14/16
Math  238-­‐001  (Test  1)       1.1,1.2,1.3,1.4,2.2,2.3,2.6   1.1  Defini▯ons   •  Differen▯al  Equa▯on-­‐  an  equa▯on  that  contains  some   deriva▯ves  of  an  unknown  func▯on   •  The  deriva▯ve  of  one  variable  (dependent  variable)  with   respect  to  another  (independent  variable)   •  Order-­‐  order  of  the  highest-­‐order  deriva▯ves  present  in   equa▯on   •  Ordinary  DE-­‐  involving  only  ordinary  deriva▯ves  with   respect  to  a  single  independent  variable.   •  Par▯al  DE-­‐  involving  only  ordinary  deriva▯ves  with  respect   to  a  more  than  one  independent  variable.   •  Linear  DE-­‐  the  dependent  variable  (y)  &  its  deriva▯ves   appear  in  addi▯ve  combina▯ons  of  their  1  powers;   depends  only  on  independent  variable  (x).   •  Nonlinear  DE-­‐  anything  that  breaks  rules  of  linear  DE   1.2  Defini▯ons   •  General  solu▯on-­‐  most  general  form  that  the  solu▯on  can   take  and  doesn’t  take  any  ini▯al  condi▯ons  into  account.   •  Actual  solu▯on-­‐  specific  solu▯on  that  not  only  sa▯sfies  the   differen▯al  equa▯on,  but  also  sa▯sfies  the  given  ini▯al   condi▯on(s).   •  An  explicit  solu▯on  is  any  solu▯on  that  is  given  in  the  form   is  once  on  the  le▯  side  and  only  raised  to  the  first  power.    An   implicit  solu▯on  is  any  solu▯on  that  isn’t  in  explicit  form.     Note  that  it  is  possible  to  have  either  general  implicit/explicit   solu▯ons  and  actual  implicit/explicit  solu▯ons.   •  Ini▯al  Value  Problem-­‐  a  differen▯al  equa▯on  along  with  an   appropriate  number  of  ini▯al  condi▯ons.     Existence  and  Uniqueness  Theorems   for  First-­‐Order  ODE’s     1.3  Defini▯ons   •  Direc▯on  Field-­‐  a  plot  of  short  line  segments   drawn  at  various  points  in  the  xy-­‐plane   showing  the  slope  of  the  solu▯on  curve  for   DE.   •  Isocline-­‐  for  DE  is  a  set  of  points  in  the  xy-­‐ plane  where  all  the  solu▯ons  have  the  same   slope  dy/dx;  thus,  it  is  a  level  curve  for  the   func▯on  f(x,y).     1.4  Defini▯ons   •  In  mathema▯cs  and  computa▯onal  science,  the  Euler  method  is  a   first-­‐order  numerical  procedure  for  solving  ordinary  differen▯al   equa▯ons  (ODEs)  with  a  given  ini▯al  value.  It  is  the  most  basic   explicit  method  for  numerical  integra▯on  of  ordinary  differen▯al   equa▯ons.   •  Summary  of  Euler's  Method   In  order  to  use  Euler's  Method  to  generate  a  numerical  solu▯on  to  an   ini▯al  value  problem  of  the  form:                yʹo=f(x, o                y(x )  =  y we  decide  upon  what  interval,  star▯ng  at  the  ini▯al  condi▯on,  we   desire  to  find  the  solu▯on.  We  chop  this  interval  into  small   subdivisions  of  length  h.  Then,  using  the  ini▯al  condi▯on  as  our  star▯ng   point,  we  generate  the  rest  of  the  solu▯on  by  using  the  itera▯ve   formulas:          =  x  +  h      =  y  +  h  f(x       n+1 n n+1 n n n to  find  the  coordinates  of  the  points  in  our  numerical  solu▯on.  We   terminate  this  process  when  we  have  reached  the  right  end  of  the   Chapter  1  Summary   •  The  order  of  a  DE  is  the  order  of  the  highest  deriva▯ve   present.  The  subject  of  this  text  is  ordinary  DE,  which   involves  deriva▯ves  with  respect  to  a  single   independent  v.     •  An  explicit  solu▯on  of  a  DE  is  a  func▯on  of  the   independent  v.  that  sa▯sfies  the  eq.  on  some  interval.   An  implicit  solu▯on  is  a  rela▯on  between  dependent   and  independent  variables  that  implicitly  defines  a   func▯on  that  is  an  explicit  solu▯on.  A  DE  typically  has   infinitely  many  solu▯ons.     •  The  conglomerate  of  specifying  direc▯ons  (slopes)  at   points  on  the  plane  is  the  direc▯on  field.    


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