Q2 (20 points) dx? Consider the 2nd-order ODE fy dy + (1 + x) + 2y = 0. dx (a) A Frobenius series solution yı(x) takes t

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Q2 20 Points Dx Consider The 2nd Order Ode Fy Dy 1 X 2y 0 Dx A A Frobenius Series Solution Yi X Takes T 1 (32.47 KiB) Viewed 27 times

Q2 (20 points) dx? Consider the 2nd-order ODE fy dy + (1 + x) + 2y = 0. dx (a) A Frobenius series solution yı(x) takes the form yı(x) = axukte Σ with ao 0 and C E R. Compute the indicial equation for c. Determine a recurrence relationship for the ak and thus deduce an explicit expression for an in terms of do. (b) A second linearly independent solution to the ODE can be written as y2(x) = yı(x) log x + 3 bxx**c. ka Show that bk = ax : + Σ k+1 for some y, that you should determine. You may use any theoretical results from the lectures as long as they are stated clearly. (c) Suppose solutions y(x) of the ODE are bounded at x = 0 and satisfy = 1, dx" x-0 for some positive integer n. Determine y(0) in this case.