University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2020
Views:

379

Pages:

1
Author:

exhibit4ej

Method: “Brute Force Technique” This method, about an ordinary point x0 , allows us to directly compute the coefficients of a power series solution up to any order. It relies on our ability to compute the nth derivative at x0 from the differential equation. ∞ X From the ansatz: y(x) = cn (x − x0 )n , compare to the Taylor expansion n=0 ∞ X y (n) (x0 ) (x − x0 )n . This tells us that we can compute the nth of y(x) = n! n=0 coefficient of the power series by: y (n) (x0 ) y 00 (x0 ) , . . . , cn = ,... 2 n! EXAMPLE: Compute the power series solution up to order 5 for the differential equation y 00 − 2xy 0 − 2y = 0, y(1) = 1, y 0 (1) = 1. Use the differential equation to compute the values of the derivatives: c0 = y(x0 ), c1 = y 0 (x0 ), c2 = y 00 (x) = 2xy 0 (x) + 2y(x) ⇒ y 00 (1) = 2(1) + 2(1) = 4 Differentiate to get a formula for y 000 (x): y 000 = 2y 0 + 2xy 00 + 2y 0 = 4y 0 + 2xy 00 ⇒ y 000 (1) = 4(1) + 2(1)(4) = 12 Continue up to y (5) (1): y (4) = 4y 00 + 2y 00 + 2xy 000 = 6y 00 + 2xy 000 ⇒ y (4) (1) = 6(4) + 2(1)(12) = 48 y (5) = 6y 000 + 2y 000 + 2xy (4) = 8y 000 + 2xy (4) ⇒ y (5) (1) = 8(12) + 2(1)(48) = 192 The solution up to 5th order is now: 4 12 48 192 y(x) = 1 + 1(x − 1) + (x − 1)2 + (x − 1)3 + (x − 1)4 + (x − 1)5 + h.o.t. 2 3! 4! 5! or simplified, 8 y(x) = 1 + (x − 1) + 2(x − 1)2 + 2(x − 1)3 + 2(x − 1)4 + (x − 1)5 + h.o.t. 5 In Maple, this can be done as: Order:=6; de:=diff(y(x),x$2)-2*x*diff(y(x),x)-2*y(x)=0; dsolve({de,y(1)=1,D(y)(1)=1},y(x),type=series); We can also analyze the relationship between the coefficients: First, note that: y (n+1) (1) = 2ny (n−1) + 2y n Substitute: y (n) = n!cn : (n + 1)!cn+1 = 2n (n − 1)!cn−1 + 2n!cn ⇒ cn+1 = This is the recurrence relation for the coefficients. 1 2 (cn + cn−1 ) n+1

Differential Equations Method “Brute Force Technique”

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