Exercise 9. Consider Legendre's differential equation (1-x)y - 2xy + y = 0, -1 < x...

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Exercise 9. Consider Legendre's differential equation (1-x²)y" - 2xy + y = 0, -1 < x < 1. a. Show that a = 0 is an ordinary point of (1). (1) b. Give a lower bound on the radius of convergence of the power series solutions to (1) of the form - Cana". n=0 y = c. Find the recursion relation satisfied by the coefficients an of the solutions y of part b. d. In applications one often takes μ = m(m + 1), m E No. Show that in this case (1) has a polynomial solution of degree m. Appropriately normalized, these are the Legendre polynomials. Exercise 9. Consider Legendre's differential equation (1-x²)y" - 2xy + y = 0, -1 < x < 1. a. Show that a = 0 is an ordinary point of (1). (1) b. Give a lower bound on the radius of convergence of the power series solutions to (1) of the form - Cana". n=0 y = c. Find the recursion relation satisfied by the coefficients an of the solutions y of part b. d. In applications one often takes μ = m(m + 1), m E No. Show that in this case (1) has a polynomial solution of degree m. Appropriately normalized, these are the Legendre polynomials.

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