MAP 2302 – INTROD TO  DIFF EQUATIONS                   FLORIDA INT'L UNIV.
      Review for Test #3   (Sept. 9th, 2008)                                                        FALL  2008
      

      REMEMBER TO BRING AN 8’’x11”  BLUE EXAM BOOKLET FOR THE TEST
  

             MAIN PROBLEM SOLVING TECHNIQUES:

1.      How to solve Cauchy-Euler ODEs by putting x = et   [i.e.,  t = ln(x) ] .

2.     How to find the indicial equation of a Cauchy-Euler ODE :
(a)  distinct real roots   (b) complex roots   (c) repeated real or complex roots.  

       3    How to find the Power Series solution of ODEs near an ordinary point .
       4.      How to find the indicial equation of linear ODEs  near a regualr singular point.
       5.   How to find the Series solution of linear ODEs near a regular singular point:
             (a) r1-r2  is not an integer     (b)  r1-r2  is zero       (a) r1-r2  is a positive integer.
             Here r1 and r2  are the roots of the indicial equation and  r1 > r2  by convention

6.     How to find the Laplace transform of  functions of exponential order.   

7.      How to solve linear homogeneous & linear non-homogenous ODEs with constant
coefficients and given intial conditions by using the Laplace transforms.

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       8.   How to find inverse Laplace transforms by using the Convolution Theorem.
       9.      How to solve linear non-homogeneous ODEs with constant coefficients and given
             intial conditions which have discontinuous RHS by using the Laplace transforms.         


              MAIN DEFINITIONS:

        Cauchy-Euler ODE, Linear second-order ODEs with variable coefficients,  Ordinary point,
        Power Series Method,  Recurrence formulas,  Regular singular point, Irregular singular point,
        Frobenius Method, Indicial Equation, Associated Cauchy-Euler ODE,  Laplace transform,
        Inverse Laplace transform,  Partial fraction method for finding inverse Laplace transforms,
        Laplace transforms of discontinuous functions, The Delta generalized-function. 
       

             MAIN THEOREMS:
       1.     The theorems on Cauchy-Euler Ordinary Differential Equations when we have :
            (a)  distinct real roots   (b) complex roots   (c) repeated real or complex roots.

2.     The theorem on Power Series solutions near ordinary points of a linear ODE.

       3    The theorems on the Frobenius Method near regular singular points when:
             (a) r1-r2  is not an integer      (b)  r1-r2  is zero       (a) r1-r2  is a positive integer.          
       4.   If  L{f} = L{g} and f and g are continuous functions on  (0, infinity) then f = g.
       5.   L{f(n)(t)} = sn.L{f(t)} -  sn-1.f(0) -  sn-2.f (1)(0) -  .  .  .  -   s.f (n-2)(0) -  f (n-1)(0).
             So  L{ f / (t)} = s.L{f(t)} -  f (0)  and  L{ f / /(t)} = s2.L{f(t)} - s.f (0) -  f /(0).
       6.   L{f*g} =  L{f}.L{g} where   (f*g)(t) =  integral of  [f(u).g(t-u)]du  from 0 to t .