note that equal to zero we have for setting the

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Unformatted text preview: is a regular singular point. The indicial equation is given by , that is, , with roots . Let . . Substitution into the ODE results in . After adjusting the indices in the first series, we obtain . Setting the coefficients equal to zero, it follows that for , . So for , ________________________________________________________________________ page 234 —————————————————————————— CHAPTER 5. —— With , one solution is . For a second solution, set Substituting into the ODE, we obtain . . Since , it follows that . More specifically, . Equating the coefficients, we obtain the system of equations Solving these equations for the coefficients, , , . Therefore a second solution is 14 . Here , therefore , , and Both of these functions are analytic at is a regular singular point. Note that The indicial equation is given by ________________________________________________________________________ page 235 —————————————————————————— CHAPTER 5. —— , that is, , with roots and . In order to find the solution corresponding to . , set Upon substitution into the ODE, we have . After adjusting the indices in the first two series, and expanding the exponential function, . Equating the coefficients, we obtain the system of equations Setting , , solution of the system results in . Therefore one solution is , , , . The exponents differ by an integer. So for a second solution, set . Substituting into the ODE, we obtain . Since , it follows that . More specifically, ________________________________________________________________________ page 236 —————————————————————————— CHAPTER 5. —— . Equating the coefficients, we obtain the system of equations Solving these equations for the coefficients, equations, set . Then , Therefore a second solution is 15 . Note the . In order to solve the remaining , , and . Furthermore, It follows that and lim lim and therefore is a regular singular point. The indicial equation is given by , that is, , with roots and . In order to find the solution corresponding to . , set Upon substitution into the ODE, we have ________________________________________________________________________ page 237 —————————————————————————— CHAPTER 5. —— . After adjusting the indices, it follows that . That is, Setting the coefficients equal to zero, we have , and for , . If we assign , then we obtain Hence one solution is , , , . . The exponents differ by an integer. So for a second solution, set . Substituting into the ODE, we obtain , since . It follows that . Now Substituting for , the right hand side of the ODE is ________________________________________________________________________ page 238 —————————————————————————— CHAPTER 5. —— . Equating the coefficients, we obtain the system of equations We f...
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