Question
Use the Frobenius Theorem to find two power series solutionis for each equation. Write at least...
Use the Frobenius Theorem to find two power series solutionis for each equation. Write at least first THREE terms for each series solution.
2 2,
Answers
"4xy"" + 1/2 y' + y = 0 rightarrow (1) Let y = sigma^infinity_n = 0 c_n I^n + r implies y' = sigma^infinity_n = 0 (n + r) c_n x^n + r-1 implies y"" = sigma^infinity_n = 0 (n + r) (n + r - 1) c_n x^n + r-2 therefore (1) implies sigma^infinity_n = 0 4(n + r) (n + r - 1) (n + r - 1) c_n x^n + r-1 + sigma^infinity_n = 0 1/2 (n + r) c_n x^n + r-1 + sigma^infinity_n = 0 c_n x^n + r = 0 implies sigma^infinity_n = 0 {4(n + r) (n + r - 1) c_n + 1/2 (n + r) c_n} x^n + r-1 + sigma^infinity_n = 0 c_n x^n + r = 0 implies sigma^infinity_n = -1 {4(n + r + 1) (n + r) c_n + 1/2 (n + r + 1) c_n + 1 + 1/2(n + r + 1) c_n + 1 + c_n] x^n + r = 0 implies 4r(r - 1) + 1/2r = 0, (n + r + 1) (4(n + r) + 1/2)c_n + 1 + c_n = 0, Forall n greaterthanorequalto 0 implies 4r^2 - 4r + 1/2r = 0, c_n + 1 - -c_n/(n + r + 1) (4(n + r) + 1/2), Forall n greaterthanorequalto 0 . . "
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