Topics

31

Term Test 1 / Problem 1 (noon)

« on: October 23, 2019, 05:54:36 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(2y+y^2\sin(x)\bigr) + \bigl(\sin(2x)+2y\cos(x)\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{4})=\sqrt{2}$.

32

Term Test 1 / Problem 1 (morning)

« on: October 23, 2019, 05:53:12 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(-y\sin(x)+y^3\cos(x)\bigr) + \bigl(3\cos(x)+5y^2\sin(x)\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{4})=\sqrt{2}$.

33

Term Test 1 / Problem 1 (main sitting)

« on: October 23, 2019, 05:51:05 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(y +3 y^2e^{2x}\bigr) + \bigl(1+2ye^{2x}\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(0)=1$.

34

Term Test 1 / Please post solutions

« on: October 20, 2019, 06:25:41 PM »

You may post solutions to Test 1. All solutions posted before this announcement were removed (all of them seem to be  prepared in advance).
One user who was too smart for his/her own good (posted several times something like a single digit to stake out the lot) got a posting ban.

Please type sin, cos, log, ...  as \sin, \cos , \log , ... to produce $\sin (x)$,... (upright and with a proper horizontal spacing)

35

Technical Questions / emoji

« on: December 20, 2018, 08:37:39 PM »

😈 👿 👹 👺 💀 👻 👽 🤖 💩 😺 😸 😹 😻 😼 😽 🙀 😿 😾

36

Final Exam / FE-P6

« on: December 18, 2018, 06:22:02 AM »

Calculate for real $n>1$
$$
I:= \int_0^\infty\frac{dx}{1+x^n}.
$$

Hint:  Consider
$$
\int_\gamma \frac{dz}{1+z^n}
$$
with with an arc of radius $R\to \infty$ and an angle $\alpha=\frac{2\pi}{n}$. Express the integral over the second straight segment through integral over the first one.

37

Final Exam / FE-P5

« on: December 18, 2018, 06:18:35 AM »

Consider $P(z)= z^3 +2z -3-i$ and, using the argument theorem and Rouché's theorem calculate the number of its roots in each of the following domains:

(a)  $\{z\colon |z-1|<1\}$;

(b)  $\{z\colon |z-1|>1, |z|<2\}$,

(c) $\{z\colon |z|>2\}$.

38

Final Exam / FE-P4

« on: December 18, 2018, 06:17:32 AM »

(a) Find the Möbius transformation (fractional-linear transformation) $f(z)$ mapping the unit disk $\{z\colon |z|<1\}$ onto itself, such that $f(0)=\frac{1}{2}$ and $f(1)=-1$.

(b) Find the fixed points of $f$ (points s.t. $f(z)=z$)

(c) Find the stretch ($|f'(z)|$) and the rotation angle ($\arg(f'(z))$) of $f$ at $z$.

39

Final Exam / FE-P3

« on: December 18, 2018, 06:14:31 AM »

Find all singular points, classify them, and find residues at these points of
$$
f(z)= \tan (z) + z\cot^2(z);
$$
infinity included.

40

Final Exam / FE-P2

« on: December 18, 2018, 06:13:19 AM »

(a) Check that circles $\{z\colon |z|=r\}$ (with $0<r<1$) are mapped onto confocal ellipses
$\{w=u+iv\colon \frac{u^2}{a^2}+\frac{v^2}{b^2}=1\}$ with $a^2-b^2=1$ and find $a=a(r)$ and $b=b(r)$.

(b) Check that segments $\{z\colon z= e^{i\theta}r,\ r\in (-1,1)\}$  are mapped onto confocal hyperbolas
$\{w=u+iv\colon \frac{u^2}{A^2}-\frac{v^2}{B^2}=1\}$ with $A^2+B^2=1$ and find $A=A(\theta)$ and $B=B(\theta)$.

(c) Find to what domain this  function  maps the unit disk  $\mathbb{D}=\{z\colon |z|<1\}$.

(d) Draw both domains.

(e) Check if the correspondence is one-to-one.

41

Final Exam / FE-P1

« on: December 18, 2018, 06:11:23 AM »

(a) Decompose into Taylor series at $0$ function $$f(z)=\frac{1}{z^2+2z+2}.$$ Find the radius of convergence $r$. Determine if the series is converging at $|z|=r$ (consider all points $z$ satisfying $|z|=r$).

(b) Decompose into Laurent's series at $\infty$ the same function. Also find the radius $R$ (so it converges as $|z|> R$).
 Determine if the series is converging at $|z|=R$ (consider all points $R$ satisfying $|z|=R$).


Hint: Represent $f(z)$ as the sum of functions of the form $\frac{a}{b+z}$.

42

Final Exam / FE-P6

« on: December 14, 2018, 08:06:54 AM »

Typed solutions only. Upload only one picture (a general phase portrait; for general one can use computer generated)
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'  = 2y(x^2+y^2+4)\, , \\
&y'  = -2x (x^2+y^2-16)
\end{aligned}\right.
\end{equation*}

(a) Find stationary points.

(b) Linearize the system at stationary points and sketch the phase portrait of this linear system.

(c) Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.

(d)  Sketch the full phase portrait.

Hint: avoid redundancy: asymptotically (un)stable node, unstable node, stable center

43

Final Exam / FE-P5

« on: December 14, 2018, 08:03:41 AM »

Typed solutions only. Upload only pictures (at all stationary points on one picture and a general phase portrait  on another; for general one can use computer generated)

For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'  = x(3x +2y -30)\, , \\
&y'  = y(2y-x-6)\,.
\end{aligned}\right.
\end{equation*}

(a) Describe the locations of all critical points.

(b) Classify their types (including whatever relevant: stability, orientation, etc.).

(c)  Sketch the phase portraits near the critical points.

(d)   Sketch the full phase portrait of this system of ODEs.

Hint: avoid redundancy: asymptotically (un)stable node, unstable node, stable center

44

Final Exam / FE-P4

« on: December 14, 2018, 07:55:46 AM »

Typed solutions only. No uploads

Find the general solution $(x(t),y(t))$ of the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x' = x-2y + \sec(t)\, &&-\frac{\pi}{2}<t<\frac{\pi}{2},\\
&y' = x -\ \,y  \,.
\end{aligned}\right.
\end{equation*}
Hint: $\sec(t)=\frac{1}{\cos(t)}$.

45

Final Exam / FE-P3

« on: December 14, 2018, 07:54:21 AM »

Typed solutions only. No uploads

Find the general solution of
\begin{equation*}
y'''-2y'' -y '+2y = \frac{12e^{2t}}{e^t+1}.
\end{equation*}
Hint: All roots are integers (or complex integers).