Topics

91

Term Test 2 / TT2 Problem 3

« on: November 24, 2018, 04:29:17 AM »

Find all singular points of
$$
f(z)=z^2(z^2-\pi^2)\cot^2(z)
$$
and determine their types (removable, pole (in which case what is it's order), essential singularity, not isolated singularity, branching point).     

In particular, determine singularity at $\infty$ (what kind of singularity we get at $w=0$ for $g(w)=f(1/w)$?).

92

Term Test 2 / TT2 Problem 2

« on: November 24, 2018, 04:28:29 AM »

(a) Find the decomposition into power series at ${z=0}$ of $f(z)=(1-z)^{-\frac{1}{2}}$. What is the radius of convergence?

(b) Plugging in $z^2$ instead of $z$ and integrating, obtain a decomposition at $z=0$ of  $\arcsin (z)$.

93

Term Test 2 / TT2 Problem 1

« on: November 24, 2018, 04:27:17 AM »

Using Cauchy's integral formula calculate
$$
\int_\Gamma \frac{dz}{z^2-2z+10},
$$
where $\Gamma$ is a counter-clockwise oriented simple contour, not passing through eiter
of $1\pm 3i$ in the following cases

(a) The point $1+3i$ is inside  $\Gamma$ and $1-3i$ is outside  it;

(b)  The point $1-3i$ is inside  $\Gamma$ and $1+3i$ is outside it;

(c) Both points $1\pm 3i$ are inside  $\Gamma$.

94

Term Test 2 / No double-dipping!

« on: November 20, 2018, 06:00:29 AM »

Do not post solutions to more than one parallel problems for different sittings, f.e. TT2-P1 and TT2B-P1.

On the other hand, you may post solution to TT2-P1  and  participate in the discussion for TT2A-P1 and TT2B-P1.

95

Term Test 2 / TT2B-P4

« on: November 20, 2018, 05:56:49 AM »

(a) Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
-3 &-2\\
\hphantom{-}5 &-5\end{pmatrix}\mathbf{x}.$$
(b)  Sketch trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

96

Term Test 2 / TT2B-P3

« on: November 20, 2018, 05:56:08 AM »

(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} -1 &\hphantom{-}1\\
-1 &-3\end{pmatrix}\mathbf{x}.$$

(b) Sketch corresponding trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

(c) Solve
$$
\mathbf{x}'=\begin{pmatrix} -1 & \hphantom{-}1\\
-1 &-3\end{pmatrix}\mathbf{x} +
\begin{pmatrix} \frac{e^{-2t}}{t^2+1}\\
0\end{pmatrix},\qquad
\mathbf{x}(0)=\begin{pmatrix} 0 \\
0\end{pmatrix}.
$$

97

Term Test 2 / TT2B-P2

« on: November 20, 2018, 05:55:03 AM »

Consider equation
\begin{equation}
y'''-y''  +4y'-4y= 8\cos(2t).
\label{2-1}
\end{equation}

(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.

(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).

(c) Find the general solution of (\ref{2-1}).

98

Term Test 2 / TT2B-P1

« on: November 20, 2018, 05:54:05 AM »

(a) Find the general solution of
\begin{equation*}
y''-5y'+6y= \frac{6e^{4t}}{e^{2t}+1}.
\end{equation*}

(b) Find solution, such that $y(0)=0$, $y'(0)=0$.

99

Term Test 2 / TT2A-P4

« on: November 20, 2018, 05:52:56 AM »

(a) Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
\hphantom{-}1 & \hphantom{-}2\\
-5 &-1\end{pmatrix}\mathbf{x}.$$
(b) Sketch trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

100

Term Test 2 / TT2A-P3

« on: November 20, 2018, 05:52:12 AM »

(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} \ 4 & \ 1\\
-3 &0\end{pmatrix}\mathbf{x}.$$

(b) Sketch corresponding trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

(c) Solve
$$
\mathbf{x}'=\begin{pmatrix}\hphantom{-}4 & \ 1\\
-3 &0\end{pmatrix}\mathbf{x} +
\begin{pmatrix} \hphantom{-}\frac{4e^{4t}}{e^t+1} \\
-\frac{4e^{4t}}{e^t+1}\end{pmatrix},\qquad
\mathbf{x}(0)=\begin{pmatrix}-1 \\
\hphantom{-}3\end{pmatrix}.
$$

101

Term Test 2 / TT2A-P2

« on: November 20, 2018, 05:51:00 AM »

Consider equation
\begin{equation}
y'''-2y''  -y'+2y= 8e^{t}.
\label{2-1}
\end{equation}

(a)
Write a differential equation for the Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.

(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).

(c) Find the general solution of (\ref{2-1}).

102

Term Test 2 / TT2A-P1

« on: November 20, 2018, 05:47:57 AM »

(a) Find the general solution of
\begin{equation*}
y''-4y=2\tanh (t).
\end{equation*}

(b) Find solution, such that $y(0)=0$, $y'(0)=0$.

103

Term Test 2 / TT2-P4

« on: November 20, 2018, 05:46:55 AM »

(a) Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
\hphantom{-}5 & \hphantom{-}5\\
-5 &-1\end{pmatrix}\mathbf{x}.$$
(b) Sketch trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

104

Term Test 2 / TT2-P3

« on: November 20, 2018, 05:46:11 AM »

(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix}\hphantom{-}2 &\hphantom{-}1\\
-3 &-2\end{pmatrix}\mathbf{x}.$$

(b) Sketch corresponding trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

(c) Solve
$$
\mathbf{x}'=\begin{pmatrix}\hphantom{-}2 &\hphantom{-}1\\
-3 &-2\end{pmatrix}\mathbf{x} +
\begin{pmatrix}\hphantom{-} \frac{4}{e^t+e^{-t}} \\
-\frac{12}{e^t+e^{-t}}\end{pmatrix},\qquad
\mathbf{x}(0)=\begin{pmatrix} 0 \\
0\end{pmatrix}.
$$

105

Term Test 2 / TT2-P2

« on: November 20, 2018, 05:45:04 AM »

Consider equation
\begin{equation}
y'''+y''  -y'-y= 8e^{-t}.
\label{2-1}
\end{equation}
(a)  Write a differential equation for the Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.

(b)  Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).

(c) Find the general solution of (\ref{2-1}).