ODE Math E1210 Midterm Exam 1 Solution
October 6, 2014
Problem 1 (6 pts)
Find the general solutions of the given dierential equations( 2 pts each).
1.
dy
dt
+ et y = t. The integrating factor u(t) satises
u = et u
t
and we could choose u = ee . We have
t
Practice Midterm Exam
1. Let y1 , y2 be two solutions of the equation y + p(t)y + q(t)y = 0 on an interval I and assume
that both p and q are continuous on I. State Abels Theorem for the Wronskian determinant
W [y1 , y2 ](t) between the two solutions y1 a
ODE Practice Midterm Exam
October 2nd, 2014
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
20 pts.
20 pts.
20 pts.
20 pts
ODE Practice Midterm Exam
October 2nd, 2014
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
20 pts.
20 pts.
20 pts.
20 pts
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0 Power series solution to regular singular point
0 Radius of convergence for power series
Suppose we have functions 19(95) and q(x) such that the indicial equation
r(7" — 1) +p(0)r + (1(0) 2 0
has a repeated root 7’ 2 T1. Then to ﬁnd we power series solu
Math E1210 Midterm Exam 2
Nov 12, 2014 '
NAME (please print): [
UNI: I 1 SCHOOL: .
Write clearly and show all work.
This is a closed book test and no calculators are alloWed.
I Problem 1 6 pts.
Problem 2 4 pts.
Problem 3 5 pts.
Problem 4 4
ODE practice problems for nal
Here is a list of topics covered after midterm 2 till the end of the class
1. Linear system with two dierent eigenvalues. Ch 7.5: 3 and 4.
2. Linear system with complex dierent eigenvalues. Ch 7.6: 3 and 4.
3. Linear system w
Name:
ID #:
Practice Midterm Exam
1. Solve the equation y + k 2 x2 y = 0 by means of a power series around x0 = 0.
Solution: You look for a power series solution y(x) =
an xn . Since,
n(n 1)an xn2 =
y (x) =
n=0
we have
n=0
n(n 1)an xn2
n=2
y + k 2 x2 y =
Practice Midterm Exam
1. Let y1 , y2 be two solutions of the equation y + p(t)y + q(t)y = 0 on an interval I and assume
that both p and q are continuous on I. State Abels Theorem for the Wronskian determinant
W [y1 , y2 ](t) between the two solutions y1 a
Name:
ID #:
Practice Midterm Exam
1. Solve the equation y + k 2 x2 y = 0 by means of a power series around x0 = 0.
2. (a) Dene the Wronskian determinant W (t) = W [y1 , y2 ](t) between two solutions y1 and y2 of
a linear homogenous second order equation y
ODE Practice nal exam
December 21, 2015
NAME (please print):
UNI:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
6
7
8
15
Problem 1 (20 pts)
Find the general solutions of the given differential equations.
1. y” + y = 8".
2. y” — 33/ + 23/ 2 cost.
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n
\/ + V: o
\ N l‘ . I . H — if -
Cwmdemﬁn eqvwmtn Y 4‘ l~ Q ~ ‘5 /\ ~ ~/\
UQAw/xl Solv’cigns U
ODE Practice nal exam
December 17, 2012
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
6
7
8
15 p
ODE Practice nal exam
December 15, 2014
NAME (please print):
ID:
SCHOOL:
Write clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Problem
Total
1
2
3
4
5
6
7
8
15 p
ODE Midterm Exam 2
Nov 18th, 2015
NAME (please print):
UNI:
SCHOOL:
VVrite clearly and show all work.
This is a closed book test and no calculators are allowed.
Problem
Problem
Problem
Problem
Problem
Tota.!
1
2
3
4
5
5 pts.
6 pts.
,1 pts.
5 pts.
5 pts.
2
Analysis and Optimization, HW 6
Due October 26
1. Let y = f (x) = ex + sin x 1.
(i) Does f (x) have the inverse function near x = 0?
(ii) Find the Taylor expansion of the inverse function of f (x) near (x, y) = (0, 0) (compute at least the first three coe
Analysis and Optimization, HW 5
Due October 19
1.
(i) Show directly that
1
x1 x2 (x1 + x2 )
2
(1)
for x1 , x2 > 0.
(ii) Using Jensens inequality, show that
n
x1 x2 xn
1
(x1 + x2 + + xn )
n
(2)
for x1 , , xn > 0.
2. Let f (x) be defined on a convex set S
Analysis and Optimization, HW 1
Due September 14
x2
x4
+
2x. Find the optimal points, the
4
2
local min/max, and the global min/max on the given domain.
1. Let f : (4, 4] R be given by f (x) =
2. Find the global min of the function f (x) = xx on (0, ). (
Analysis and Optimization, HW 7
Due November 2
1. Use Lagrange mutipliers to find the maximum and minimum values of the function
subject to the given constraint.
(i) f (x, y) = exy ; x3 + y 3 = 16
(ii) f (x, y, z) = xyz; x2 + 2y 2 + 3z 2 = 6
(iii) f (x, y