Practice Problems: Midterm Exam
1. Find the general solution to each of the following differential equations.
(a) y = 1 + e 2 x
(b) y = x + xy 2
(c) xy + 3y = x 2
(d) y + (xy)2 = 0
(e) y = ex 2y
(f) ln(y ) + 2 ln(x) = y
2. Given that y = 8x 2 + 10y 2 and
Math 213
Name:
Quiz 5
1. [10 points] The following matrix species a linear system involving the variables x1 , x2 and x3 .
2
1 2
2
0
3 3
3
2
4 3 3
1 1
1 5
(a) Use elementary row operations to transform this matrix to reduced row echelon form.
(b) Use y
Math 213
Name:
Midterm Exam
1. [6 points] Find a constant k so that y = x k is a solution to the equation x 2 y = 20y.
2. [6 points] Use guess & check to nd one solution to the differential equation e 4x y = y 3 + e 6x .
3. [6 points] Find the general sol
Math 213
Name:
Quiz 2
1. [6 points] Find the general solution to the following differential equation.
y = xe x+y
2. [6 points] Find the general solution to the following differential equation.
xy 3y = x5 .
3. [8 points] Solve the following initial value p
Math 213
Name:
Quiz 3
1. [5 points] Consider the following initial-value problem.
y = x + y2 3.6,
y(0) = 2.
Use Eulers method with step size x = 0.5 to estimate y(1).
2. [3 points] In a certain electrical circuit with an inductor and a resistor, the elect
Math 213
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Quiz 1
1. Find the general solution to the equation
1
y = + sin(2x).
x
2. Find all solutions to the equation
y 8y + 12y = 0
of the form y = eAx , where A is a constant.
3. Solve the following initial value problem:
y = y,
y(0) = 10,
y (0) =
Practice Problems: Final Exam
1. Find a basis for the subspace of R4 spanned by the following vectors:
1
4
1
2
5
1
5
4
5
7
,
,
,
,
.
1
4
1
2
6
1
6
7
8
9
2. Express the matrix
4 1
0 1
as a linear combination of the matrices
1 1
1 1
Math 213
Name:
Quiz 6
1. [10 points] Find the inverse of the following matrix.
1 3 0
2 7 1
1 3 1
2. [7 points] Compute the following determinant.
1 2 0
6 8 3
4 2 1
3. [3 points] Determine the parity of the following permutation. You must show your work
Math 213
Name:
Quiz 9
1. [8 points] Determine which of the following subsets are subspaces. Your answer to each of these
questions should be yes (if it is a subspace) or no (if its not a subspace).
(a) The subset of R2 consisting of all vectors of the for
Math 213
Name:
Quiz 10
1. [8 points] Find the general solution to the following differential equation
y 3y + 4y 12y = 0.
Your nal answer should not involve any complex numbers.
2. [10 points] Find a basis for the = 3 eigenspace of the following matrix.
4
Practice Problems: Midterm Exam
1. Find the general solution to each of the following differential equations.
(a) y = 1 + e 2 x
(b) y = x + xy 2
(c) xy + 3y = x 2
(d) y + (xy)2 = 0
(e) y = ex 2y
(f) ln(y ) + 2 ln(x) = y
2. Given that y = 8x 2 + 10y 2 and
Math 213
Name:
Quiz 8
1. [10 points] Find a basis for the subspace of R4 spanned by the following vectors:
v1 = (1, 3, 4, 1),
v2 = (2, 6, 8, 2),
v3 = (1, 4, 5, 3),
and
v4 = (2, 3, 5, 4).
2. [5 points] Find a matrix with three rows whose null space has dim
Math 213
Name:
Quiz 7
1. [10 points] Find a basis for the null space of the following matrix.
1 3 2 4 1
2 6 3 5 5
3 9 4 6 9
2. [10 points] Determine whether the following vectors are linearly independent or linearly dependent
in R4 . If they are linearl
Practice Problems: Final Exam
1. Find a basis for the subspace of R4 spanned by the following vectors:
1
4
1
2
5
1
5
4
5
7
,
,
,
,
.
1
4
1
2
6
1
6
7
8
9
2. Express the matrix
4 1
0 1
as a linear combination of the matrices
1 1
1 1
Math 213
Name:
Quiz 4
1. Find the values of x and y that satisfy the following matrix equation:
1 0
3 x 0
2 0 =
y 0 2
2 2
2. Find a 2 2 skew-symmetric matrix A such that
A2 = 4I,
where I denotes the 2 2 identity matrix.
y 0
x 4