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Unformatted text preview: Mathematics 238 test three due Friday, May 27, 2005 1. Given the diﬀerential equation (x2 + 4)y + 3xy + 4y = 0 • list the singular points of the equation. • for which x-values will a power series solution of the form
∞ ak (x−3)k converge?
k=0 2. For which values of x does the each of following power series converge? (a)
∞ (x − 4)2k k 9k k=1 ∞ (b)
k=1 a2k x2k where a2 = 1 & ka2k+2 = 3(k + 1)a2k for k ≥ 1 3. Find a power series solution about x = 0 (through the degree four term) to the ﬁrst-order initial value problem dy = x2 + y 2 dx extra credit parts of problem (3) • compare your solution to a numerically obtained solution from a math software package or website. (plot the two solutions together and compare) • compare your solution to the ﬁrst few Picard iterates. 4. Given the second-order diﬀerential equation y + xy + y = 0 • for which values of x will power series solutions of this diﬀerential equation converge? • ﬁnd the recursion relation for the power series solution about x = 0. • write the general solution through the degree ﬁve terms. • ﬁnd the solution (through degree ﬁve terms) to the initial value problem y (0) = 0 and y (0) = 1. • estimate the accuracy of your solution of the previous part when −0.5 ≤ x ≤ 0.5. and y (0) = 1 Mathematics 238 test three, part two due Monday, June 6, 2005 5. Use the elimination method to solve the system of equations d2 x1 dt2 d2 x2 dt2 = x2 − x1 − 1 = −(x2 − x1 − 1) + (x3 − x2 − 1) d2 x
3 dt2 = −(x3 − x2 − 1) Begin the process by rewriting the system in the format (D2 + 1)x1 − x2 = −1 −x1 + (D2 + 2)x2 − x3 = 0 −x2 + (D2 + 1)x3 = 1 ...
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This note was uploaded on 11/01/2010 for the course ENGINEERIN PHYS222 taught by Professor Ewqfqw during the Spring '09 term at University of Washington.
- Spring '09