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Homework Math # 9
Due Nov. 29
Texas A & M University
Fall 2011
1. Without actually solving the following dierential equations, nd a lower
bound for the radius of convergence of power series solutions about the ordinary point x = 0. Also about the ordinary point x = 1.
a ) [1 points] (x2 25)y + 2xy + y = 0
b ) [1 points] (x2 2x + 10)y + xy 4y = 0
2. Use the codes HW9P1aModel.m, HW9P1aModelSolver, HW9P1bModel.m
and HW9P1bModelSolver.m available at the course webpage to solve dierential equations of Problem 1 numerically.
a ) [4 points] Describe your observations when x values pass through the
singular points or exceed the radius of convergence.
b ) [2 points] Does the positive or negative initial values play a role in
the behavior of solutions? why?
c ) [4 points] Include a few graphs to support your statements.
3. For each of the following dierential equations, nd two linearly independent power series solutions about the ordinary point x = 0 .
a ) [8 points] y 2xy + y = 0
b ) [8 points] (x 1)y + y = 0
c ) [10 points] (x2 + 2)y + 3xy y = 0
4. [10 points] Use the power series method to solve the initial value problem
(x 1)y xy + y = 0, y (0) = 2, y (0) = Determine 6.
5. the singular points of each dierential equation. Classify each
singular point as regular or irregular.
a ) [2 points] (x2 9)2 y + (x + 3)y + 2y = 0
1
1
y=0
b ) [2 points] y y +
x
(x 1)3
1
c ) [2 points] x2 (x 5)2 y + 4xy + (x2 25)y = 0
6. For each of the following dierential equations: (i) show that the indicial roots of the singularity do not dier by an integer, (ii) use the method
of Frobenius to obtain two linearly independent series solutions about the
regular singular point x = 0, (iii) form the general solution on (0, ).
a ) [8 points] 2xy y + 2y = 0
b ) [10 points] 4xy + .5y + y = 0
c ) [10 points] 3xy + (2 x)y y = 0
7. The system of dierential equations for the currents i2 (t) and i3 (t) in an
electrical network shown in Figure 1 is
d
dt
i2
i3
=
R1 /L1
R1 /L1
R1 /L2 (R1 + R2 )/L2
i2
i3
+
E /L1
E/L2
a ) [7 points] Use the method of undetermined coecients to solve the
system if R1 = 2, R2 = 3, L1 = 1h, L2 = 1h, E = 60V, i2 (0) = 0, and
i3 (0) = 0.
b ) [3 points] Determine the current i1 (t).
Figure 1: Network in Problem 7
8. [8 points] Use variation of parameters to solve the given system.
X=
2 1
4
2
X+
2
sin 2t
2 cos 2t
e2t