Page 280 - Schaum's Outline of Differential Equations
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CHAP.  27]     LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS              263



         SOLUTIONS AROUND THE ORIGIN OF HOMOGENEOUS EQUATIONS
            Equation  (27.7) is homogeneous when g (x)  = 0, in which case Eq.  (27.2)  specializes  to



         Theorem 27.1.  If x = 0 is an ordinary point of Eq. (27.3), then the general  solution in an interval  containing
                       this point has the form





                       where a 0 and  a^  are arbitrary constants and Ji(x)  and y 2(x)  are linearly independent  functions
                       analytic at x = 0.

            To  evaluate  the  coefficients  a n  in  the  solution  furnished by  Theorem  27.1,  use  the  following  five-step
         procedure known as the power series method.
         Step  1.  Substitute into the left  side of the homogeneous  differential equation the power series







                together with the power series for





                and




         Step 2.  Collect powers of x and set the coefficients of each power of x equal to zero.
         Step 3.  The  equation obtained by setting the coefficient  of x"  to zero in Step 2 will contain - terms for a  finite
                                                                                  a ;
                                                    a- term having the largest subscript. The resulting equation
                number of j  values. Solve this equation for the ;
                is known as the recurrence formula  for the given differential  equation.
         Step 4.  Use the recurrence formula to sequentially determine a, (j  = 2, 3, 4,...) in terms of a 0 and a^.

         Step 5.  Substitute the coefficients determined  in  Step 4 into Eq.  (27.5) and rewrite the solution in  the form
                ofEq.  (27.4).
            The  power  series  method  is  only  applicable  when  x = 0  is  an  ordinary  point.  Although  a differential
         equation must be in the form of Eq. (27.2) to determine whether x'  = 0 is an ordinary point, once this condition
         is  verified,  the power series method  can be used on either form  (27.7) or  (27.2).  If P(x) or  Q(x)  in  (27.2)  are
         quotients  of  polynomials,  it  is  often  simpler  first  to  multiply  through  by  the  lowest  common  denominator,
         thereby  clearing  fractions, and then to apply the power series method  to the resulting equation in  the form of
         Eq. (27.7).



         SOLUTIONS AROUND THE ORIGIN OF NONHOMOGENEOUS EQUATIONS
            If  (f>  (x) in  Eq.  (27.2) is analytic at x = 0, it has  a Taylor series expansion around that point and  the power
         series method given above can be modified to solve either Eq.  (27.7) or (27.2). In  Step  1, Eqs. (27.5) through
         (27.7)  are  substituted into  the left  side of the nonhomogeneous  equation;  the right  side is  written as a Taylor
         series around the origin.  Steps 2 and 3 change  so that the coefficients of each power of x on the left  side of the
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