Math 220, Midterm I
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(last)
Signature:
The following rules apply:
There are a total of 20 points on this 50 minutes exam. This contains 6 pages (including
this cover page) and 4 problems. Check to see if any page is missing. Enter all
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M220 F13 Review and extra practice problems for Final
Final Schedule: 1:30 pm4:00 pm Thursday, December 19, 2013
1
First order odes: y = f (t, y ).
Main Topics:
A rough classication of dierential equations (ODE or PDE, order, linear or nonlinear);
Exist
Math 220 F13 #16887 Extra Practice Problems for Exam 2
Coverage: 3.1 3.8
1. Solve the nonhomogeneous equations using the method of undetermined coecients:
(i) y + y 2y = 3et
(ii) y + y 2y = 3et
(iii) y y = sin t + 2t.
2. (i) Use the method of undetermine
Math 220 F13 #16887 Review Problems for Exam 1
Coverage: All material covered from 1.1 3.1 except 2.7, 2.8, 2.9
1. Determine the order and linearity of each ODE:
(i) y + et yy + cos y = sin t
(ii) y + (sin t)y t2 y = tet
(iii) (ln x)
d5 y
d2 y
2 = ex y
d
Math 220 #16887 Exam 2 Nov. 7, 2013
Solution Key
1. Find the solution of the initial value problem
y 4y + 8y = 0,
y (0) = 1, y (0) = 2.
Solution.
A general solution yc (t) of the equation
The characteristic equation r2 4r + 8 = 0 has roots r = 2 2i.
Th
Math 220 F2013 Project 1 (Due on Thursday, 10/17/2013)
dy
I. (Page 49) The 1st order ODE
= f (x, y ) is called homogeneous if f (x, y ) = g (y/x)
dx
for some function g . This type of ODEs can be reduced to separable ones as follows:
Introduce a new func
Math 220 #16887 Exam 1 September 26, 2013
Solution Key
1. Determine the order and linearity of each ODE:
(i) (t2 + 1)y + 2et y 2t sin y = ln(t2 + 1).
(ii) (cos t)y = (tan t)y (t2 + 1)y .
(iii) y + t(y )4 y = 1 + t2 .
Answer: 2nd order, nonlinear
Answer: 3
Math 220 Fall 2013 Project 2 (Due Tuesday 12/05/13)
Solve the following initial value problems using Laplace Transform:
1. y 4y + 3y = te2t ;
y (0) = 0, y (0) = 1.
2. y + 6y + 10y = (t 1);
3. y + 4y = g (t) where g (t) =
y (0) = 0, y (0) = 0.
1,
sin(t), t
Practice Test (nal)
Outline of the course
1. Order and linearities.
2. First order dierential equation
(a) y + p(t)y = q (t)
(b) Separable equation:
dy
M (t)
=
dt
N (y )
(c) Exact equation.
(d) y = f (y ): analysis of stability of solution.
(e) Domain of
This is an old exam. The purpose of this is to show you the format of the actual midterm exam.
The problems in the actual midterm exam will be DIFFERENT from those in this old exam.
The midterm exam # 1 will be held during 1:002:15PM, Thursday, October 3
Math 220: Review of Chapters 1, 2, and 8
Chapter 1: Introduction
Basic concepts: DE, order, linearity, homogeneity, solution, the general solution, integral
curve.
Direction eld, method of isoclines.
Chapter 2: 1st order DEs
dy
dt
= f (t, y )
Linear eq
Math 220, Midterm I
Name (Print): (rst)
(last)
Signature:
The following rules apply:
There are a total of 20 points on this 50 minutes exam. This contains 6 pages (including
this cover page) and 4 problems. Check to see if any page is missing. Enter all
Math 220: Review of Chapter 6: The Laplace transform
I. Denitions
1. The Laplace transform and inverse Laplace transform.
2. The unit step functions.
II. Properties and formulae
1. Formulae on Page 319 of the textbook.
III. Techniques
1. Find the inverse
This is an old exam. The purpose is to show you the format of the actual nal exam. The
problems in the actual nal exam will be DIFFERENT from those in this old exam.
The nal exam will be held from 1:30PM to 4:00PM, Tuesday, December 17, 2013
in Snow Hall
HW 12 due Nov 26, Monday
1. Find the general solution to the nonhomogeneous linear system
(a) x =
2 1
3 2
x+
2
0
r1 = 1,
1
r2 = 1,
1
1
=
2
x(1) =
,
1
3
=
,
et et
et 3et
1 3et et
1 (t) =
t
et
2 e
3et
1 (t)g (t) =
et
1
1
et
1
3
x(2) =
et
(t) =
u(t) =
u1 (t
HW 10 due Nov 12, Mon
1. Transform given equations into system of rst order linear equations.
(a) u + 0.5u + 2u = 0.
Let x1 = u,
x1
x2
x2 = x1
0
1
2 0.5
x1
x2
(b) t2 u + tu + (t2 0.25)u = sin(t)
1
t2 0.25
sin t
u + u+
=2
2
t
t
t
Let x1 = u, x2 = x1
0
1
x