MATH 1B Lecture 32: More Power Series Solutions to Differential Equations
Document Summary
Math 1b: calculus - lecture 32: more power series solutions to differential equations. Q: find the first 3 nonzero terms for y"" + xy" + y = 0, given that y(0) = 0 and y"(0) = 1. A: we have (d2y/dx2) + x(dy/dx) + y = 0, and y(0) = 0 and y"(0) = 1. Put y = the sum from n=0 to of (anxn). Therefore, (dy/dx) = the sum from n=1 to of [(n+1)(an+1)xn] = the sum from n=0 to of [(n+1)(an+1)(xn)]. Also, (d2y/dx2) = the sum from n=1 to of [(n+1)(n)(an+1)(xn-1)]. = the sum from n=0 to of [(n+2)(n+1)(an+2)(xn)]. {the sum from n=0 to of [(n+2)(n+1)(a. We require (d2y/dx2) + x(dy/dx) + y = 0. So {the sum from n=0 to of [(n+2)(n+1)(an+2)(xn)]} + x{the sum from n=0 to of [(n+1)(an+1) (xn)]} + {the sum from n=0 to of (anxn)} = 0. + {the sum from n=0 to of (anxn)} =0.
