MATHEMATICS 3161: Fall 2013
Assignment # 2 (Due Date: Oct. 4)
1. Find all singular points of the given equation and determine whether each one is regular or
irregular.
a) (1 x2 )2 y + x(1 x)y + (1 + x)y = 0
b) (x2 x 2)y + (x + 1)y + 2y = 0,
c) x2 y + 2(ex
MATHEMATICS 3161: Fall 2013
Assignment # 3 (Due Date: Oct. 21)
1. Transform the given equation or initial value problem into a system of rst order equations
or a system of rst order equations with initial conditions.
a) u(5) + tu(3) + (t 1)u = 0,
b) u + 0
MATHEMATICS 3161: Fall 2013
Assignment # 1 (Due Date: Sept. 23)
1. Determine the radius of convergence of the given power series.
a)
n=1
(2x + 1)n
n2
b)
n=1
(x x0 )n
n
2. Rewrite the given expression as a sum whose generic term involves xn .
nan xn1 +
a)
MATHEMATICS 3161: Fall 2013
Assignment # 2 (Due Date: Oct. 7)
1. Find all singular points of the given equation and determine whether each one is regular or
irregular.
a) (1 x2 )2 y + x(1 x)y + (1 + x)y = 0
b) (x2 x 2)y + (x + 1)y + 2y = 0,
c) x2 y + 2(ex
MATHEMATICS 3161: Fall 2013
Assignment # 3 (Due Date: Oct. 21)
1. Transform the given equation or initial value problem into a system of rst order equations
or a system of rst order equations with initial conditions.
a) u(5) + tu(3) + (t 1)u = 0,
b) u + 0
MATHEMATICS 3161: Fall 2013
Assignment # 4 (Due Date: Nov. 8)
5
2e3t
, y2 ( t ) =
are known to be solutions of the homogeneous
1
e3t
linear system y = Ay , where A is a real 2 2 constant matrix,
a) Verify that the two solutions form a fundamental solution
MATHEMATICS 3161: Fall 2013
Assignment # 5 (Due Date: Nov. 29)
1. Consider the initial value problem
y = ty,
y (0) = 1 ,
hand-calculate the rst ve iterations of Eulers method, the rst three iterations of Heuns
method and the rst two iterations of Runge-Ku
MATHEMATICS 3161: Fall 2013
Assignment # 1 (Due Date: Sept. 23)
1. Determine the radius of convergence of the given power series.
a)
n=1
(2x + 1)n
n2
n=1
b)
(x x0 )n
n
Solution:
(2x + 1)n+1
n2
(n + 1)2
= |2x + 1|,
a) nlim
= nlim |2x + 1|
n
(2x + 1)
(n + 1
MATHEMATICS 3161: Fall 2013
Assignment # 5 (Due Date: Nov. 29)
1. Consider the initial value problem
y = ty,
y (0) = 1 ,
hand-calculate the rst ve iterations of Eulers method, the rst three iterations of Heuns
method and the rst two iterations of Runge-Ku
MATHEMATICS 3161:
Fall 2013
Midterm Exam (Oct. 18, 2013)
n 1 n 2
n+1
1n
n+1 n
1n 1
1
(
x
+
x=
x+
x= +
+ 2 ) xn
2
2
3 n=1 n + 3 n
n=2 n + 1
n=1 n
n=0 n + 3
n=1 n
1. Sol:
5x
, x = 0 is not a singular point, and both functions are
1 x2
rational, so x = 0 is
MATHEMATICS 3161: Fall 2013
Assignment # 4 (Due Date: Nov. 8)
5
2e3t
, y2 (t) =
are known to be solutions of the homogeneous
1
e3t
linear system y = Ay , where A is a real 2 2 constant matrix,
1. The functions y1 (t) =
a) Verify that the two solutions for