2. Consider the 2nd-order ODE day dy + (1+x) + 2y = 0. d.x2 dx (a) A Frobenius series solution yı(x) takes the form yı(x

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2 Consider The 2nd Order Ode Day Dy 1 X 2y 0 D X2 Dx A A Frobenius Series Solution Yi X Takes The Form Yi X 1 (213 KiB) Viewed 33 times

2. Consider the 2nd-order ODE day dy + (1+x) + 2y = 0. d.x2 dx (a) A Frobenius series solution yı(x) takes the form yı(x) = axxk+c, (3 k=0 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 ak in terms of ao. (b) A second linearly independent solution to the ODE can be written as y2(x) = yı (x) log x + bkxk+c. k=1 Show that k br = ak Σ Vjt 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 day = 1, dxn 20 for some positive integer n. Determine y(0) in this case.