Download test three

Mathematics 238 test three
due Friday, May 27, 2005
1. Given the differential equation
(x2 + 4)y 00 + 3xy 0 + 4y = 0
• list the singular points of the equation.
∞
• for which x-values will a power series solution of the form X a (x−3)k converge?
k
k=0
2. For which values of x does the each of following power series converge?
(a)
∞
X
(x − 4)2k
k=1
(b)
∞
X
k9k
a2k x2k where a2 = 1 & ka2k+2 = 3(k + 1)a2k for k ≥ 1
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
and y(0) = 1
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 00 + xy 0 + 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 (0) = 1.
• estimate the accuracy of your solution of the previous part when −0.5 ≤ x ≤ 0.5.
Mathematics 238 test three, part two
due Monday, June 6, 2005
5. Use the elimination method to solve the system of equations
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d2 x1
dt2
d2 x2
dt2
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= x2 − x1 − 1
= −(x2 − x1 − 1) + (x3 − x2 − 1) 
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d2 x
3
dt2
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= −(x3 − x2 − 1)
Begin the process by rewriting the system in the format
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(D2 + 1)x1 − x2 = −1
−x1 + (D2 + 2)x2 − x3 = 0
−x2 + (D2 + 1)x3 = 1
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