MATH 2270 Lecture : Week 9.pdf
21 Dec 2013
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5. 3 more details on power series solutions to odes. A(x)y(cid:48)(cid:48) + b(x)y(cid:48) + c(x)y = 0. (5. 38) Any value of x for which a(x) (cid:54)= 0 is called an ordinary point. If a(x ) = 0, however, then x is said to be a singular point. For example, for the equation x2y(cid:48)(cid:48) + (2 x2)y = 0 (5. 39) has a singular point at x = 0. For series solutions about singular points, we require techniques that can be quite di cult; in this course, we will concern ourselves only with nding series solutions about ordinary points. Find the power series solution to to at least the x4 term. First, we let y(cid:48)(cid:48) + 2y(cid:48) xy = 0. (cid:88) n=0 y = anxn, di erentiate twice, and then sub into the equation: (cid:88) n=0 n(n 1)anxn 2 + 2 nanxn 1 n=2 n=1 n=0 anxn+1 = 0. (5. 43)
