Topics

106

Term Test 2 / TT2-P1

« on: November 20, 2018, 05:43:50 AM »

(a) Find the general solution of
\begin{equation*}
y''+4y=2\tan (t),\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}.
\end{equation*}

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

107

Quiz-5 / Q5 TUT 0501

« on: November 18, 2018, 04:21:35 AM »

It looks like I missed it

Transform the given system into a single equation of second order and find the solution $(x_1(t),x_2(t))$, satisfying initial conditions
$$\left\{\begin{aligned}
& x'_1= -0.5x_1 + 2x_2, &&x_1(0) = -2,\\
&x'_2= -2x_1 - 0.5x_2, &&x_2(0) = 2
\end{aligned}\right.$$

108

Quiz-6 / Q6 TUT 5301

« on: November 17, 2018, 04:14:52 PM »

Locate each of the isolated singularities of the given function $f(z)$ and tell whether it is a removable singularity, a pole, or an essential singularity.
If the singularity is removable, give the value of the function at the point; if the singularity is a pole, give the order of the pole:
$$
f(z) =\pi \cot(\pi z).
$$

109

Quiz-6 / Q6 TUT 5201

« on: November 17, 2018, 04:12:56 PM »

Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{1}{1-\cos(z)};\qquad z_0=0\quad\text{(four terms of the Laurent series)} .
$$

110

Quiz-6 / Q6 TUT 5101

« on: November 17, 2018, 04:12:01 PM »

Locate each of the isolated singularities of the given function $f(z)$ and tell whether it is a removable singularity, a pole, or an essential singularity.

If the singularity is removable, give the value of the function at the point; if the singularity is a pole, give the order of the pole:
$$
f(z)= \frac{e^z-1}{e^{2z}-1}.
$$

111

Quiz-6 / Q6 TUT 0301

« on: November 17, 2018, 04:11:06 PM »

Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{\sin(z)}{(z-\pi)^2};\qquad z_0=\pi.
$$

112

Quiz-6 / Q6 TUT 0203

« on: November 17, 2018, 04:10:26 PM »

$\newcommand{\Res}{\operatorname{Res}}$
If $f$ is analytic in $\{z\colon 0< |z - z_0| < R\}$ and has a pole of order $l$ at $z_0$ , show that
$$
\Res \bigl(\frac{f'}{f}; z_0\bigr)=-l.
$$

113

Quiz-6 / Q6 TUT 0202

« on: November 17, 2018, 04:09:19 PM »

Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{1}{e^z-1};\qquad z_0=0\quad \text{(four terms of the Laurent series)} .
$$

114

Quiz-6 / Q6 TUT 0201

« on: November 17, 2018, 04:08:42 PM »

Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{z^2}{z^2-1};\qquad z_0=1.
$$

115

Quiz-6 / Q6 TUT 0102

« on: November 17, 2018, 04:08:01 PM »

Find the Laurent series for the given function $f(z)$ about the indicated point. Also, give the residue of the function at the point.
$$
f(z)=\frac{z}{\sin^2(z)};\qquad z_0=0\quad \text{(four terms of the Laurent series)} .
$$

116

Quiz-6 / Q6 TUT 0101

« on: November 17, 2018, 04:07:17 PM »

$\newcommand{\Res}{\operatorname{Res}}$
If $f$ is analytic in $\{z\colon |z - z_0| < R\}$ and has a zero of order $m$ at $z_0$ , show that
$$
\Res \bigl(\frac{f'}{f}; z_0\bigr)=m.
$$

117

Quiz-6 / Q6 TUT 5102

« on: November 17, 2018, 04:01:01 PM »

The coefficient matrix contains a parameter $\alpha$.

(a) Determine the eigenvalues in terms of $\alpha$.
(b)  Find the critical value or values of  $\alpha$  where the qualitative nature of the phase portrait for the system changes.
(c) Draw a phase portrait for a value of  $\alpha$ slightly below, and for another value slightly above, each critical value.
$$\mathbf{x}' =\begin{pmatrix}
4 &\alpha\\
8 &-6
\end{pmatrix}\mathbf{x}.$$

118

Quiz-6 / Q6 TUT 5101

« on: November 17, 2018, 03:59:47 PM »

The coefficient matrix contains a parameter $\alpha$ . In each of these problems:

(a) Determine the eigenvalues in terms of $\alpha$.
(b)  Find the critical value or values of  $\alpha$  where the qualitative nature of the phase portrait for the system changes.
(c) Draw a phase portrait for a value of  $\alpha$ slightly below, and for another value slightly above, each critical value.
$$\mathbf{x}' =\begin{pmatrix}
2 &-5\\
\alpha & -2
\end{pmatrix}\mathbf{x}.$$

119

Quiz-6 / Q6 TUT 0801

« on: November 17, 2018, 03:58:12 PM »

Find the general solution of the given system of equations:
$$\mathbf{x}'=
\begin{pmatrix}
1 &1 &1\\
2 &1 &-1\\
-8 &-5 &-3
\end{pmatrix}\mathbf{x}.$$

120

Quiz-6 / Q6 TUT 0701

« on: November 17, 2018, 03:57:36 PM »

Find the general solution of the given system of equations:
$$\mathbf{x}'=
\begin{pmatrix}
3 &2 &4\\
2 &0 &2\\
4 &2 &3
\end{pmatrix}\mathbf{x}.$$