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 echelon form (Gauss-Jordan elimination) and solve the
sys
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 y1 + c2 y2 which satises the initial conditions
y (0) =
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
= ex .
Multiplying the dierential equation by (x) yields
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.
Solution: We integrate with respect to x to get
y ( x) =
1
(
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 = 6e2x + 6e2x = 0,
so the dierential equation is satis
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 Wednesday, March 7.
1. E/P 3.4, Problem 2
Determine the period
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, March 2.
1. E/P 3.1, Problem 6
Verify that y1 = e2x an
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, February 24.
1. E/P 1.5, Problem 4
Find the general so
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, February 17.
1. E/P 1.2, Problem 2
Find a function y =
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, February 13.
1. E/P 1.1, Problem 2
Verify by substitut
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 the dierential equation
dy
= y cos x.
dx
1
[5 points]
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 coordinates of
5
with respect to B .
4
[3 points] 1b. Find th
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 value problem
xy + 3y = 5x2 ,
2
y (2) = 5.
3. Solve the
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 N/m.
Solution: The dierential equation is
0.75x + 48x =
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
ans: 3.5:2
r2 r 2 = 0
r = 1, 2
3.
t
yH = c1 e
40.34
2t
+
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
= 2xdx
y
dy
=
y
(1 + x)dx
1
ln |1 + y| = x + x2 + C
2
2xd
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 initial value problems:
dy
= y 1/3 ;
dx
dy
= y 1/3 ;
(b)
dx
d
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) directly by nding a zero linear combination with nonzero
c
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 at point (x, y) is the sum
of x and y.
r
r
(b) The lin
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 ).
ans:
(3.4:2)
Determine whether the following vectors form
M341 H7 (S. Zhang) 1.2 1.3.
1.
52.20
2.
52.22
(1.2:1(a-f),)
Determine if it is in row echelon form, or
reduced row echelon form.
1
0
2
0
3
1
4
2
1
0
0
0
0
0
0
0
1
3
0
0
(1.2:2(a-c),)
Find the consistence. For those having a
unique solution, nd the solutio
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 = c1 et c2 et
x
y
ans:
et
et
= c1
+ c2
et
et
x =y
6.
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 death rates and are both pro
portional to 1/ P . If P0 = 9
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) dierential equation:
ans: 1.1:28
3y = 2y
(a) The tangent li