2r2 + r3 r3 , r2 + r4 r4 ,
2 1
2
0
M341 Study Guide for E 3 (S. Zhang) .
1. Find the row echelon form (okay to have non-one diagonals,
Gauss elimination) and solve the system. Find the reduced
row ec
Math 341 - Gewecke
Homework 4 Solutions
1. E/P 3.1, Problem 6
Verify that y1 = e2x and y2 = e3x are solutions of the dierential equation
y + y 6y = 0.
Then nd a particular solution of the form y = c1
Math 341 - Gewecke
Homework 2 Solutions
1. E/P 1.5, Problem 4
Find the general solution of the dierential equation
2
y 2xy = ex .
Solution: First, we calculate the integrating factor
(x) = e
2x dx
2
=
Math 341 - Gewecke
Homework 2 Solutions
1. E/P 1.2, Problem 2
Find a function y = f (x) satisfying the given dierential equation and the prescribed initial condition.
dy
= (x 2)2 ,
dx
y (2) = 1.
Solut
Math 341 - Gewecke
Homework 1 Solutions
1. E/P 1.1, Problem 2
Verify by substitution that y = 3e2x is a solution of the dierential equation
y + 2y = 0.
Solution: Since
y = 6e2x ,
we have
6e2x + 2 3e2x
Math 341 - Gewecke
Homework 5
Instructions: You may work with other students on these problems, but each student must submit their
own work. This assignment is due at the beginning of class on Wednesd
Math 341 - Gewecke
Homework 4
Instructions: You may work with other students on these problems, but each student must submit their
own work. This assignment is due at the beginning of class on Friday,
Math 341 - Gewecke
Homework 3
Instructions: You may work with other students on these problems, but each student must submit their
own work. This assignment is due at the beginning of class on Friday,
Math 341 - Gewecke
Homework 2
Instructions: You may work with other students on these problems, but each student must submit their
own work. This assignment is due at the beginning of class on Friday,
Math 341 - Gewecke
Homework 1
Instructions: You may work with other students on these problems, but each student must submit their
own work. This assignment is due at the beginning of class on Monday,
Math 341 Section 011: Final Exam
NAME:
This test has 12 questions on 12 pages, plus a blank page at the end.
The points per page are 5,5,5,5,6,6,6,6,6,6,6,6.
[5 points] 1. Find the general solution of
Math 341 Section 011: Test #3
This test has 6 questions on 6 pages. Each page is worth the same.
1. Let B be the ordered basis cfw_u1 , u2 where u1 =
3
4
and u2 =
.
2
3
[3 points] 1a. Find the coordi
Math 341 Section 011: Test #2
This test has 6 questions on 6 pages. Each question is worth the same.
1. Solve the system
1
0
0
1
1
0
1
1
1
0
1
0
0
0
1
1
0
1
0
1
0
x1
0
x2 0
1
1 x3 = 0
1 x4 0
0
x
Math 341 Section 011: Test #1
This test has 6 questions on 6 pages. Each question is worth the same.
1. Find the general solution of the dierential equation
dy
+ 6xy 2 = 0.
dx
1
2. Solve the initial v
Math 341 - Gewecke
Homework 5 Solutions
1. E/P 3.4, Problem 2
Determine the period and frequency of the simple harmonic motion of a body of mass 0.75 kg on the end
of a spring with spring constant 48
M341 H5 (S. Zhang) 3.5-6.
1.
yp =
1
3
1
+
cos 2x
sin 2x
2 26
13
Find the general solution:
40.30
y = yH + y p =
y y 2y = 3x + 4
3t
3t
+ c2 sin
)
2
2
3
1
1
cos 2x
sin 2x
+ +
2 26
13
= et/2 (c1 cos
a
M341 H2 (S. Zhang) [b].
4.
(1.4:17.)
Find the general solution of
8.7
1.
(1.4:1.)
dy
= (1 + x)(1 + y)
dx
Find the general solution of
8.2
dy
+ 2xy = 0
dx
ans:
dy
=
1+y
ans: Separable equation!
dy
=
What we know:
M341 Study Guide for E Final (S. Zhang) .
1.
6.6
Determine if the existence and uniqueness theorem for the
rst order DE guarantees the solution (in a neighborhood)
for each of the initia
M341 Study Guide for E 2 (quiz 4, due Wed) (S.
Zhang) .
We use the integration by parts method, letting u = (x +
1), dw = ex , du = dx, w = ex
1. Show the linearly dependence (1) by Wronskian, (2) dir
M341 Study Guide for E 1 (S. Zhang) .
1. Find a dierential equation y = f (x, y) so that a solution y = g(x) has the described geometric property for its
graph.
x
r40
T
(a) The slope of the graph of g
M341 H12 (S. Zhang) EP 5.2: 2, 6, 8, 13(no graph), 28, 29,
38
EP 5.3: 1, 3, 9.
1.
(5.2:2)
3.
(5.2:8)
Solve the system by eigenfunctions:
90.24
x1 = x1 5x2 , x2 = x1 x2
Solve the system by eigenfunctio
M341 H10 (S. Zhang) 3.4: 2, 5, 8, 10, 14
L 3.5: 1, 2, 5, 6
L 3.6: 1, 2, 4, 5, 8.
1.
68.20
(c) What is the dimension of Span(x1 , x2 , x3 )?
(d) Give a geometric description of Span(x1 , x2 , x3 ).
an
ans:
M341 H8 (S. Zhang) 1.4,2.1.
1.
56.20
1
5
3
1 1
1 2
1 1
2 1 1
r2 +r3
1 2 = U
3
2
6
4
3r1 +r2
2
2r1 +r3
(1.4:3)
For each of the following pairs of matrices, nd
an elementary matrix E such that
ans:
M341 H6 (S. Zhang) 4.1,1.1.
1.
46.24
x =y =x
x x=0
(4.1:1)
Transform the dierential equation into an
equivalent system of rst-order dierential equations.
x = c1 et + c2 et
x + 3x + 7x = t2
y = x
ans: M = 200 The birth+death rate is proportional
to the gap to the maximal population.
M341 H3 (S. Zhang) [b].
1.
22.2
(2.1:10)
Suppose that when a certain lake is stocked
with sh, the birth and dea
M341 H1 (S. Zhang) .
1.
2.9
(a) The line tangent to the graph of g at (x, y) intersects
the x-axis at the point (x/2, 0).
Find a solution of type y = erx for the (homogeneous,
constant coecient) diere