I only need parts 5 and 6
7. POWER SERIES METHODS Consider for an parameter pe R, p > 0 the ODE for the unknown function yp: (-1,1) CR + R given by (1 22) (x) x 4p(x) + p2 - 4p(x) = 0. (1) Classify all points x ER and oo into ordinary points, regular singular points and irregular singular points. In particular show that x = 0) is an ordinary point. Remark: Distinguish cases if necessary. (2) Consider the base point x = 0. Please use the power series method with the ansatz 4p(x) = { cjp.de j=0 to find a recursion relation for the coefficients Cp for j e Z, j> 0. (3) Consider the case pez, p> 0 and set q:=p. (a) Consider the case q even and denote the solution ya by M, in this case. Choose C0g = 1 and C1,9 = 0 in this case. Please show that Mq is an even polynomial of degree q (ie. all odd coefficients vanish, C4,9 +0 and Cq+2,9 = 0). (b) Consider the case q odd and denote the solution yg by N, in this case. Choose Co,q = 0 and C1,4 = 0 in this case. Please show that N, is an odd polynomial of degree q (ie. all even coefficients vanish, C4,9 70 and Cq+2,9 = 0). (4) Define for q eZ,q> 0 the polynomials Ed by (-1)} Mg(x) q is even, g(x) = {(-1)7.q.Ng(x) q is odd, and compute , , 2, 3. (5) Please show that Eq(x) = cos(q arccos(x)) for q = 0,1, 2, 3, where arccos denotes the inverse to the cosine function. Remark: Use trigonometric identities. (6) Please plot the graph of the functions , for q = 0,1,2,3 on the interval (-1,1) CR. Remark: This part may be answered with the information from part (5) independent of part (1)-(4). 7. POWER SERIES METHODS Consider for an parameter pe R, p > 0 the ODE for the unknown function yp: (-1,1) CR + R given by (1 22) (x) x 4p(x) + p2 - 4p(x) = 0. (1) Classify all points x ER and oo into ordinary points, regular singular points and irregular singular points. In particular show that x = 0) is an ordinary point. Remark: Distinguish cases if necessary. (2) Consider the base point x = 0. Please use the power series method with the ansatz 4p(x) = { cjp.de j=0 to find a recursion relation for the coefficients Cp for j e Z, j> 0. (3) Consider the case pez, p> 0 and set q:=p. (a) Consider the case q even and denote the solution ya by M, in this case. Choose C0g = 1 and C1,9 = 0 in this case. Please show that Mq is an even polynomial of degree q (ie. all odd coefficients vanish, C4,9 +0 and Cq+2,9 = 0). (b) Consider the case q odd and denote the solution yg by N, in this case. Choose Co,q = 0 and C1,4 = 0 in this case. Please show that N, is an odd polynomial of degree q (ie. all even coefficients vanish, C4,9 70 and Cq+2,9 = 0). (4) Define for q eZ,q> 0 the polynomials Ed by (-1)} Mg(x) q is even, g(x) = {(-1)7.q.Ng(x) q is odd, and compute , , 2, 3. (5) Please show that Eq(x) = cos(q arccos(x)) for q = 0,1, 2, 3, where arccos denotes the inverse to the cosine function. Remark: Use trigonometric identities. (6) Please plot the graph of the functions , for q = 0,1,2,3 on the interval (-1,1) CR. Remark: This part may be answered with the information from part (5) independent of part (1)-(4)
