MATH 257 Midterm: MATH 257+316 2015 Winter Test 1 Formula
Document Summary
Math 257-316 pde formula sheet - nal exam. 2 (1 cos(2t)) cosh2 t sinh2 t = 1. 2 (cosh(2t) 1) sinh2 t = 1. Series solutions for y + p(x)y + q(x)y = 0 ( ) around x = x0. Ordinary point x0: two linearly independent solutions of the form: y(x) = p n=0 an(x x0)n. Regular singular point x0: rearrange ( ) as: (x x0)2y + [(x x0)p(x)](x x0)y + [(x x0)2q(x)]y = 0. If r1 > r2 are roots of the indicial equation: The second linerly independent solution y2 is of the form: Case 1: if r1 r2 is neither 0 nor a positive integer: y2(x) = Case 2: if r1 r2 = 0: y2(x) = y1(x) ln(x x0) + Xn=1 bn(x x0)n+r2 for some b1, b2 Case 3: if r1 r2 is a positive integer: y2(x) = ay1(x) ln(x x0) + Xn=0 bn(x x0)n+r2 where b0 = 1.
