Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2013
1
Solutions to Section 1.1
TrueFalse Review:
1. FALSE. A derivative must involve some derivative of the function y = f (x), not necessarily the rst
derivative.
2. TRUE. The initial conditions accompanying a dierential equation consist of the values of y,
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
December 2016
Midterm III Review Session
Math 250B
SOLUTIONS
Name:
Problem 1. Suppose A is a square matrix such that
A3 2A2 + 4A + 5I = 0.
If is an eigenvalue of A, show that
3 22 + 4 + 5 = 0.
SOLUTION: There is some eigenvector v such that Av = v. Then
A
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
May 9, 2016
Midterm III
Math 250B
SOLUTIONS
Name:
Problem 1. (9 points) Decide whether the matrix A below is diagonalizable or not,
circle your answer, and show work to justify it. If the matrix is diagonalizable, find
an invertible matrix S and a diagona
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Problem 1. (9 points) Decide whether the matrix A below is diagonalizable or not, circle your
answer, and show work to justify it. If the matrix is diagonalizable, nd an invertible matrix S and
a diagonal matrix D such that S 1 AS = D.
5
8
16
1
8 .
A= 4
4
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
December 2016
Midterm III Review Session
Math 250B
Name:
Problem 1. Suppose A is a square matrix such that
A3 2A2 + 4A + 5I = 0.
If is an eigenvalue of A, show that
3 22 + 4 + 5 = 0.
Problem 2. Suppose A is a square matrix with eigenvalues 1 6= 2 . Show t
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Math 250B
Group Work #3
SOLUTIONS
Spring 2017
Problem 1. Solve the differential equation
(3x 2y)
dy
= 3y.
dx
SOLUTION: We see that this differential equation is homogeneous of degree zero, so we set y = xV .
Then y 0 = V + xV 0 and we have
(3x 2xV )(V + x
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Math 250B
Group Work #1
SOLUTIONS
Spring 2017
Problem 1. Find all values of r such that y(t) = ert is a solution to the differential
equation
(a): y 00 + 3y 0 10y = 0.
SOLUTION: We have y 0 = rert and y 00 = r2 ert . Substituting these expressions into th
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Math 250B
Group Work #2
SOLUTIONS
Spring 2017
Find the general solution to each differential equation below:
Problem 1:
(x + 1)
dy
=x+6
dx
x+6
SOLUTION: We use separation of variables to obtain dy = x+1
dx. Now let w = x + 1. Then
w+5
5
dw = dx, and we ha
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
MATH 250B EXAM 2 _ 6/25/152
Name: n Mania; /
Show your work! Wag' 7L _
I. (12 pts.) Determine whether the given set S of vectors is closed under addition and
closed under scalar multiplication. (Take the set of scalars to be the set of all real
numbers.)
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
MATH 2503 EXAM 2 6/25/15
Name:
Show your work!
I. (12 pts.) Determine whether the given set S of vectors is closed under addition and
closed under scalar multiplication. (Take the set of scalars to be the set of all real
numbers.) ls S a subspace of V?
(a
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
MATH 2508 WORKSHEET 10
Name:
Name:
Name:
Name:
1. Decide whether or not the given mapping T is a linear transformation. If T is a linear
transformation, nd the kernel of T and its dimension.
(a) TszH, isamappingdefinedby T(ax2+bx+c)=(a+b+c)x+l.
/(0):(O+0+
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
MATH 2508
EXAM l 6/11/152
Name:
Show your work!
(,2 x
mp=3x g f
._/_Z
@G# 71 3X
jj '1 ,x Fit 0/
x2 2_
+1f* 2. (14 pts.) Solve the given initialvalue problem
_ 4y
dx 4x 2y
y(1) =1
(homogeneous rstorder DE)
dquv
V+XZJ7't/_72\/ V V 2 1/2 2
0/
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
MATH 2508 WORKSHEET 9
Name:
Name: ._.
Name:
Name:
Show your work!
I. Find a spanning set for a real vector space V. Determine whether this spanning set of
vectors is linearly independent in V. Is this set a basis for V? What is the dimension
of V?
(a)
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Math 250B
Midterm III Information
Spring 2017
WHEN: Wednesday& Thursday, May 3&4, in class (no notes, books, calculators
I will supply a list of common annihilators.)
REVIEW SESSIONS:
Mon, May 1, 2:302:55 PM in room MH468
Tue, May 2, 4:455:25 PM in roo
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Chapter 8:
Linear Differential Equation of order .
Section 8.1
General Theory for Linear Differential Equation.
Notations:
= interval on the real line (, ) or (, )
0 () = cfw_continuous function in
() = cfw_differentiable function on I
() = cfw_ time
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
HO mew alt( SOU'ii'onf
5:9 +t"0n 435'
2. cfw_(1. l), (l, 1). These vectors are elements of R2. Since there are two vectors. and the dimension of 1R2
1 1
l1
= 0. Now
is two. Corollary 1.5.17 states that the vectors will be linearly dependent if an
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
HQMQUDOPk Sol U" tan 3
gecl't'on gal
lA s
_ 2 2 _ =0
13.det(AAI)=04=+ 2 _3_Al0+=>A+ A [5
=>(A3)(A+5)=0=>A=30rA=5.
' 2 6 v o
lfA=3 then (llAI)V=0 assumes the form [ 2 (i J [1;] =[0]
=> v1 3112 = 0. If we let ya = r E R, then the solution set of this
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
H'o manner1% gallql'i'ons
m
Static n Li". 8
3. (a). n =
(b)
5; a basis for rowspacepl) is given by 1,2, 3, l 5)
m = l; a basis for colspacel) is given by [1.
I 2 3
6. (a). n. = 3; a rowechelon form of A is 0 l 2
D 0 0
cfw_(1, 2, 3), (0, l, 2)
Conseq
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Homework Sol +20 nJ'
l 2 U
2 t 0 0 _
1. (a). T(10,5) = l 8 5 1 = 0 . Thus, x (lees belong to lxerljI).
. . J
8 16 0
l 2 l
2 4 I 2 .
(b). T(1,l) = ti 8 l = II . Thus, 3: does not belong to ixer(T).
S 16 8
l 2 (l
'2 4 2 0 .
(c). T(2, l) = 1 8 l = 0 . Th
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Homeworit gal (lions
Sachem 8.7
1. Given 3;  y' + 9y = 4133:1113: =r r2 6r + 9 = (r 3)"! = 0 =3 r' E cfw_3,3 = y,.(;z:) 2 clear +chea.
Let 111(3) = e33 and y2(;r:) = we. We have ll[yl,y2](:r) = er. Then a particular solution to the given
differentia
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Homage ark gcholcms
Seciiam 1.1:
1 . We have
Av=[:i][3][t]=2v.
as required.
1 3 l 4 1
2.Av=[2 2H1]=[d]=4[ 1 ]=Av
1 2 6 2 (i 2
3 AV 2 2 2 5 l = 3 = 3 l = Av.
2 l 8 l 3 l
l 1 (.1 +162
4. Since v = cl 0 + c; 3 = 3cz , it follows that
"3 0 3L1
l rl l
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
vu'vv IV! '5.
Instructions: Show your work and indicate your reasoning. You will not receive credit if
you do not clearly Show how you are obtaining your answers. Box your nal answer. Do all
work on the exam. Manage your time carefully. Dont spend too muc
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Math 250B
Some More Problems On Ch 8 and 9
SOLUTIONS
Problem 1. Let L := D2 x1 D + 4x2 . Verify that y(x) = sin(x2 ) is in the kernel of L.
SOLUTION: Since the operators D and D2 appear, we need to compute the first two derivatives
of y(x):
y 0 (x) = 2x c
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Math 250B
Midterm III Information
Fall 2016
SOLUTIONS TO PRACTICE PROBLEMS
Problem 1. Determine whether the following matrix is diagonalizable or
not. If it is, find an invertible matrix S and a diagonal matrix D such that
S 1 AS = D.
2
0
A :=
0
0
0
3
Intro to Differential Equations and Linear Algebra
MATH 250B

Spring 2010
Problem 2.(5 points) If A is an eigenvalue of the matrix A, Show that A 5 is an eigenvalue of
the matrix A BI.
X is eValu. ul A ._> OHM2M.) :0
Of EVE/Av
Mm 5V bath 31345 _
PVT! 517: A 1' SV
[15 539V 05y SI
. a, S (S e ,Vlhe '90! HI
Problem 3. Consider