In Problems 1 through 10, find a power series solution of the given differential equation. Determine the radius of convergence of the resulting series, and use the series in Eqs. (5) through (12) to identify the series solution in terms of familiar elementary functions.

y' = x²y

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and || COS X = sin.x = coshx = M8 M8 M8 |- || - x Σ n=0 Σ n=0 ∞0 sinh x = Σ n=0 H=0 n! = 1 + x + (-1)" x²n (2n)! (-1)"x²n+1 (2n + 1)! xan (2n)! x2n+1 (2n + 1)! = 1 00 (−1)n+1x" In(1 + x) = Σ n=1 n = 1+ (1 + x) = 1 + ax + 2! = * =X- 2! 2! 4! =-- α(α – 1)x2 2! 3! x 3 3! x2 4! + + _x" = 1 + x + x2 + x +…; n=0 για 5! 3 3 α(α − 1)(α – 2)x3 3! +... (5) (6) (7) (8) (9) (10) (11) (12)