is substituting into the ode in the product of the

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Unformatted text preview: ——————— CHAPTER 5. —— Section 5.3 2. Let be a solution of the initial value problem. First note that Differentiating twice, Given that two equations give 3. Let and , the first equation gives and . and the last be a solution of the initial value problem. First write Differentiating twice, Given that two equations give 4. Let and , the first equation gives and . and the last be a solution of the initial value problem. First note that Differentiating twice, Given that two equations give and 5. Clearly, and converge everywhere. , the first equation gives and . and the last are analytic for all . Hence the series solutions 7. The zeroes of are the three cube roots of . They all lie on the unit circle in the complex plane. So for , . For , the nearest ________________________________________________________________________ page 194 —————————————————————————— CHAPTER 5. —— root is , hence 8. The only root of is zero. Hence 9 . . . and are analytic for all . . and are analytic for all . . and are analytic for all . . The only root of is . Hence . . and are analytic for all . The zeroes of are . Hence . The zeroes of are . Hence . The zeroes of are . Hence The only root of is . Hence . . and are analytic for all . . and are analytic for all . 12. 13 The Taylor series expansion of Let , about . , is . Substituting into the ODE, First note that . The coefficient of in the product of the two series is Expanding the individual series, it follows that Setting the coefficients equal to zero, we obtain the system , general solution is , , . Hence the , ________________________________________________________________________ page 195 —————————————————————————— CHAPTER 5. —— . We find that two linearly independent solutions are Since and converge everywhere, 13. The Taylor series expansion of , about Let The coefficient of . , is . Substituting into the ODE, in the product of the two series is , in which . It follows that Expanding the product of the series, it follows that 1 Setting the coefficients equal to zero, , , Hence the general solution is , , . We find that two linearly independent solutions are ________________________________________________________________________ page 196 —————————————————————————— CHAPTER 5. —— The nearest zero of is at 14. The Taylor series expansion of Let Hence , about , is . Substituting into the ODE, The first product is the series . The second product is the series Combining the series and equating the coefficients to zero, we obtain Hence the general solution is . We find that two linearly independent solutions are The coefficient radius of convergence is analytic at , but its power series has a . ________________________________________________________________________ page 197 —————————————————————————...
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This note was uploaded on 03/11/2014 for the course MA 303 taught by Professor Staff during the Spring '08 term at Purdue.

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