A) True                     B) False

2.) If $\phi$ is a solution to a first order linear homogeneous differential equation, then $c\phi$ is also a solution to this equation.

                   A) True                     B) False

3.) If $\phi$ is a solution to a first order linear homogeneous differential equation with constant coefficients, then $c\phi$ is also a solution to this equation.

                   A) True                     B) False

1.) If $b^2 - 4ac < 0$ then the solution to the initial value problem $ay'' + by' + cy = 0$, $y(0) = -1$, $y'(0) = -3$ is complex valued.

                   A) True 657686                     B) False 657688

2.) If $b^2 - 4ac < 0$ then the solution to the initial value problem $ay'' + by' + cy = 0$, $y(0) = -1$, $y'(0) = -3$ is real valued.

                   A) True 657689                     B) False 657690

1.) $L(g) = g'' + p(t) g' + q(t)g$ is a linear function on the space of all twice differentiable functions.

                   A) True 672134             B) False 672135

2.) $L(g) = g'' + p(t) g' + q(t)gg'$ is a linear function on the space of all twice differentiable functions.

                   A) True 672136             B) False 672137

1.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(0) = 1, y'(0) = 3

                   A) True 734194             B) False 734195

2.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(0) = 1, y(1) = 3

                   A) True 734213             B) False 73214

3.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(0) = 1, y'(1) = 3

                   A) True 734215             B) False 734216

4.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(1) = 1, y'(1) = 3

                   A) True 73217             B) False 734218

1.) When taking the derivative with respect to $t$, $(y^2)' = 2y$

                   A) True 741376             B) False 741377

2.) When taking the derivative with respect to $t$, $(y^2)' = 2yy'$

                   A) True 741378             B) False 741379

3.) If $y(0) = -2$ and $y^2 = g(t)$, then $y(t) = \sqrt{g(t)}$

                   A) True 741380             B) False 741381

4.) If $y(0) = -2$ and $y^2 = g(t)$, then $y(t) = -\sqrt{g(t)}$

                   A) True 741382             B) False 741383

1.) $\Sigma_{n = 2}^\infty n(n-1)a_nx^{n-2} = \Sigma_{j = 0}^\infty (j+2)(j+1)a_{j+2}x^{j} =\Sigma_{n = 0}^\infty (n+2)(n+1)a_{n+2}x^{n}$

                   A) True 118810             B) False 118811

1. Suppose $f(x) = \Sigma a_n (x - 3)^n$ has a radius of convergence = $r$ about the $ 3$. Then we can define the domain of $f$ to be $(3- r, 3 + r)$.

               A) True                         B) False

2. Suppose $f(x) = \Sigma a_n (x + 1)^n$ has a radius of convergence = $4$ about the $ -1$. Then we can define the domain of $f$ to be $(3, 5)$.

               A) True                         B) False

3. Suppose $f(x) = \Sigma a_n (x + 1)^n$ has a radius of convergence = $4$ about the point $x_0 = -1$. Then we can define the domain of $f$ to be $(-5, 3)$.

               A) True                         B) False

1. If $p:(a, b) \rightarrow R$, $q:(a, b) \rightarrow R$, and $g:(a, b) \rightarrow R$ are continuous and $a < t_0 < b$, then there exists a unique function $y = \phi(t)$,   $\phi:(a, b) \rightarrow R$ that satisfies the initial value problem $y'' + p(t) y' + q(t)y = g(t)$,   $y(t_0) = y_0$,    $y'(t_0) = y_1$.

               A) True                         B) False

2. $D(f) = f'$ is a linear function.

               A) True                         B) False

3. The differential operator $D$ is a linear operator

               A) True                         B) False

1. If $P$ and $Q$ are polynomial functions with no common factors, then $y = Q(x)/P(x)$ is analytic at $x_0$ if and only if $P(x_0) \not= 0$. Moreover the radius of convergence of the power series for $ Q(x)/P(x) $ about the point $x_0$ is

where $||x_0-x||$ = distance from $x_0$ to $x$ in the complex plane.

               A) True                         B) False

2. The radius of convergence of the power series for $f(x) = {x \over (x^2 + 1)(x + 2)}$ about the point $x_0 = {1 \over 4}$ is $\geq \sqrt{({1 \over 4})^2 + (\pm 1)^2} = {\sqrt{17} \over 4}$

               A) True                         B) False

3.) ${x \over (x^2 + 1)(x + 2)}$ $= \Sigma_{n=0}^\infty a_n(x - {1 \over 4})^n$ where $a_n = {f^{(n)}( {1 \over 4}) \over n!}$ for all values of $x \in ({1 - \sqrt{17} \over 4}, {1 + \sqrt{17} \over 4})$.

               A) True                         B) False

1. Suppose $A \left[\matrix{5 \cr 6}\right] = \left[\matrix{5 \cr 13}\right]$,   $A \left[\matrix{3 \cr 5}\right] = \left[\matrix{9 \cr 15}\right]$,   $A \left[\matrix{-1 \cr 3}\right] = \left[\matrix{17 \cr 19}\right]$,   $A \left[\matrix{2 \cr 1}\right] = \left[\matrix{-4 \cr -2}\right]$

State the 2 eigenvalues of $A$:

State 5 eigenvectors of $A$:

1. If $P$ and $Q$ are polynomial functions with no common factors, then $y = Q(x)/P(x)$ is analytic at $x_0$ if and only if $P(x_0) \not= 0$. Moreover the radius of convergence of the power series for $ Q(x)/P(x) $ about the point $x_0$ is

where $||x_0-x||$ = distance from $x_0$ to $x$ in the complex plane.

               A) True                         B) False

2. The power series for $f(x) = -(2 + x)$ converges for all $x$.

               A) True                         B) False

3.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{4 \cr 13}\right]$ are linearly independent.

               A) True                         B) False

4.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{4 \cr 13}\right], \left[\matrix{1 \cr 1}\right]$ are linearly independent.

               A) True                         B) False

5.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{-10 \cr -12}\right]$ are linearly independent.

               A) True                         B) False

The following are all equivalent:

$v_1, v_2, ..., v_n$ are linearly independent

$c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ has a unique of solutions

There does not exists an $i$ such that $v_i$ is a linear combination of $v_1, ..., v_{i-1}, v_{i+1}, ..., v_n$.

Let $A = [v_1 v_2 ... v_n]$

$Ax = 0$ has a unique solution.

$Ax = b$ has at most one solution.

If $A$ is a square matrix, $Ax = b$ has a unique solution.

If $A$ is a square matrix, $det(A) \not= 0$

0 is not an eigenvalue of $A$.

1. If $p:(a, b) \rightarrow R$, $q:(a, b) \rightarrow R$, and $g:(a, b) \rightarrow R$ are continuous and $a < t_0 < b$, then there exists a unique function $y = \phi(t)$,   $\phi:(a, b) \rightarrow R$ that satisfies the initial value problem $y'' + p(t) y' + q(t)y = g(t)$,   $y(t_0) = y_0$,    $y'(t_0) = y_1$.

               A) True                         B) False

2. If $A$ is a matrix whose elements consist of continuous functions of $t$, then there exists a unique function ${\bf x}(t) = {\bf f}(t)$ that satisfies the initial value problem ${\bf x}' = A{\bf x}$ ,   ${\bf x}(t_0) = {\bf x_0}$,    ${\bf x}'(t_0) = {\bf x_1}$.

               A) True                         B) False

3. If $A$ is a matrix whose elements consist of continuous functions of $t$, then there exists a unique function ${\bf x}(t) = {\bf f}(t)$ that satisfies the initial value problem ${\bf x}' = A{\bf x}$ ,   ${\bf x}(t_0) = {\bf x_0}$.

               A) True                         B) False

In 22M:100, I am grading your proofs based on what you write, but not as much on how you write. In more advanced math classes how you write a proof can be as important as what your write. Thus a proof that earns you 95% in this class could earn you only 60% in a different class. In some cases, minor changes that are actually obvious once you know about them can greatly increase your grade on proofs.

To prepare for graduate school consider taking several of the following graduate courses (form study groups to read first chapter or two during the summer -- some classes start in ch 2):

22M:142:AAA also known as(MATH:5600:0AAA) Nonlinear Dynamics with Numerical Methds

22M:115 (MATH:5200) Introduction to Analysis I

22M:120 (MATH:5000) Abstract Algebra I

22M:132 (MATH:5400) General Topology

Undergraduate courses:

22M:140 (MATH:4610) Continuous Mathematical Models

Only Fall courses are listed. All the above graduate courses also have a 2nd part offered in the Spring.

For Spring: