MATH 1B Lecture 31: Power Series Solutions to Differential Equations
Document Summary
Math 1b: calculus - lecture 31: power series solutions to differential equations. There are many examples of differential equations that cannot be solved analytically given an elementary function solution. We have (dy/dx) = e-(x^2) given that y=1 when x=0. You only need to give the first three non-zero terms in a power series for y, centered at zero. Recall that ex = sum from n=0 to of (xn / n!) So e-(x^2) = the sum from n=0 to of ((-x2)n / n!) = the sum from n=0 to of [((-1)n(x2n))/n!] = 1 - x2 + (x4/2) - (x6/3! Put y = the sum from n=0 to of (anxn) So as y = 1 when x = 0, a0 = 1. Also, (dy/dx) = the sum from n = 0 to of n(an)xn-1. = the sum from n = 1 to of n(an)xn-1. = the sum from n = 0 to of [(n+1)(an+1)(xn)]
