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Answers for Part A to Part C
Consider the second-order differential equation y" – 4y= 0, y(0) = ao , y(0) =a1 a) Using the power series ansatz y (x) = P on=an Xn show the recurrence relation an+2 = 4an / (n+ 2)(n+ 1), n = 0, 1, 2, ... b) Determine power series solutions y1, y2 near xo= 0 such that y (x) =aoy1(x) +a1y2(x) using the recurrence relation (no verification necessary). c) Show that y1(x) = cosh(2x) =(e^(2x)+e^(-2x))/ 2, y2(x) =1/2 sinh(2x) =(e^2x-e-A(2x))4 using the Taylor expansions cosh(z) = .£Xn=0 z2n/ (2n)!, sinh(z) = .EXn=0 (Z2n+1)/(2n+ 1)! What is the radius of convergence of the series solutions y1, y2? (Hint: You may use the fact that the functions sinh(z), cosh(z) have no singularities in the complex plane

Part A) | 4an ((η + 2) (η + 1)αη+2 – 4αη = 0 και αη+2 = (η + 2)(η +1) Part B) y(α) = αο - 4kg2k 4kg2k+1 (2k)! +αι Σ (2k + 1)! + k=0 k=0 yı(x) Y2(c)

Part C) 91(a) = 44,22 cosh(2x) 22k x2k (2x)2k (2k)! (2k)! (2k)! k=0 k=0 k=0 4k x2k+1 1 22k+1x2k (2k + 1)! 2 (2k + 1)! k=0 1 y2(x) IM: (2x)2k+1 (2k + 1)! sinh(22). 2 k=0 k=0