9.The Laguerre13 differential equation is xy′′+(1−x)y′+λy=0.xy″+1−xy′+λy=0. a.Show that x = 0 is a regular singular point. b.Determine the indicial equation, its roots, and the recurrence relation. c.Find one solution (for x > 0). Show that if λ = m, a positive integer, this solution reduces to a polynomial. When properly normalized, this polynomial is known as the Laguerre polynomial, Lm(x).

4. as soon as possible please

5. Use the meyhod of undetermined coefficients to solve the differential equation. a) ¹y" + y + y = x² = x b) y" + 3y = -48t²e-3t

. Consider the differntial equation y" + y' = e". Which of the following statements are true? (2 marks) а. у * is a solution to the differential equation. 2 b. y = 3 is a solution to the differential equation. 2 c. y = + 3 is a solution to the differential equation. 2 d. If z y', and if z(0) = 1 then z = 2 e. The most general solution to this equation is y = + Cje-® + C2 for constants C1, C2 E R. 2