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MAT244--Misc / FE Marks
« on: April 20, 2018, 05:22:07 PM »
I removed this topic since it does not carry any useful information but also some students exposed some personal info. Not much, but better to be on the safe side.
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2
APM346--Announcements / Grading FE
« on: April 14, 2018, 10:40:19 AM »
I finished grading FE. Need to enter marks, calculate etc. Bonus (karma) will be done about 5pm. After this: no changes
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Web Bonus Problems / Exam Week
« on: April 12, 2018, 03:38:18 PM »
Consider problem:
\begin{align}
& \Delta u=0 &&\text{in }x^2+y^2<1, \ \ y>0,
\label{1}\\
& u|_{y=0}=x^2 &&\text{as }|x|<1,\label{2}\\
& u|_{x^2+y^2=1}=1,&& \text{as } y>0.
\label{3}
\end{align}
We want to separate variable $r$ and $\theta$ but the conditions as $\theta=0,\pi$ are inhomogeneous.
So we want to make them homogeneous. Find $v$, so that $u:=v$ satisfies (\ref{1}) and (\ref{2}) but not necessarily (\ref{3}), so $v$ is not unique. Can you suggest a candidate?
Then $w=u-v$ will satisfy (\ref{1}), homogeneous condition (\ref{2}), modified (\ref{3}). Find $w$ by separation, and then $u=v+w$.
\begin{align}
& \Delta u=0 &&\text{in }x^2+y^2<1, \ \ y>0,
\label{1}\\
& u|_{y=0}=x^2 &&\text{as }|x|<1,\label{2}\\
& u|_{x^2+y^2=1}=1,&& \text{as } y>0.
\label{3}
\end{align}
We want to separate variable $r$ and $\theta$ but the conditions as $\theta=0,\pi$ are inhomogeneous.
So we want to make them homogeneous. Find $v$, so that $u:=v$ satisfies (\ref{1}) and (\ref{2}) but not necessarily (\ref{3}), so $v$ is not unique. Can you suggest a candidate?
Then $w=u-v$ will satisfy (\ref{1}), homogeneous condition (\ref{2}), modified (\ref{3}). Find $w$ by separation, and then $u=v+w$.
4
MAT244--Announcements / Grading FE
« on: April 12, 2018, 01:12:30 PM »
While other instructors and TAs are grading their parts of FE, I am busy with grading FE APM346, which was 5 hours earlier. I will deal with your class next week
5
Final Exam / FE-P6
« on: April 11, 2018, 08:48:39 PM »
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = -2xy\, , \\
&y'_t = x^2+y^2-1
\end{aligned}\right.
\end{equation*}
a. Linearize the system at
stationary points and sketch the phase portrait of this linear system.
b. Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.
c. Sketch the full phase portrait.
\begin{equation*}
\left\{\begin{aligned}
&x'_t = -2xy\, , \\
&y'_t = x^2+y^2-1
\end{aligned}\right.
\end{equation*}
a. Linearize the system at
stationary points and sketch the phase portrait of this linear system.
b. Find the equation of the form $H(x,y) = C$, satisfied by the trajectories of the nonlinear system.
c. Sketch the full phase portrait.
6
Final Exam / FE-P5
« on: April 11, 2018, 08:47:26 PM »
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = x (x -y+1)\, , \\
&y'_t = y (x - 2)\,.
\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.
\begin{equation*}
\left\{\begin{aligned}
&x'_t = x (x -y+1)\, , \\
&y'_t = y (x - 2)\,.
\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.
7
Final Exam / FE-P4
« on: April 11, 2018, 08:44:51 PM »
Find the general solution of the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = \hphantom{-}x + y +\, \frac{e^{t}}{\cos(t)}\, ,\\
&y'_t = - x + y +\, \frac{e^{t}}{\sin(t)}\,.
\end{aligned}\right.
\end{equation*}
\begin{equation*}
\left\{\begin{aligned}
&x'_t = \hphantom{-}x + y +\, \frac{e^{t}}{\cos(t)}\, ,\\
&y'_t = - x + y +\, \frac{e^{t}}{\sin(t)}\,.
\end{aligned}\right.
\end{equation*}
8
Final Exam / FE-P3
« on: April 11, 2018, 08:42:26 PM »
Find the general solution of
\begin{equation*}
y''' -6y'' +11y'- 6y=2\frac{e^{3x}}{e^x+1} .
\end{equation*}
\begin{equation*}
y''' -6y'' +11y'- 6y=2\frac{e^{3x}}{e^x+1} .
\end{equation*}
9
Final Exam / FE-P2
« on: April 11, 2018, 08:40:12 PM »
Find the general solution by method of the undetermined coefficients:
\begin{equation*}
y'''-2y''+4y'-8y= 16 e^{2t} + 30\cos(t);
\end{equation*}
\begin{equation*}
y'''-2y''+4y'-8y= 16 e^{2t} + 30\cos(t);
\end{equation*}
10
Final Exam / FE-P1
« on: April 11, 2018, 08:39:30 PM »
Find the general solution
\begin{multline*}
\bigl(2xy \cos(y)-y^2\cos(x)\bigr)\,dx +
\bigl(2x^2\cos(y)-yx^2\sin(y)-3y\sin(x)-5y^3\bigr)\,dy=0\,.
\end{multline*}
Hint. Use the integrating factor.
\begin{multline*}
\bigl(2xy \cos(y)-y^2\cos(x)\bigr)\,dx +
\bigl(2x^2\cos(y)-yx^2\sin(y)-3y\sin(x)-5y^3\bigr)\,dy=0\,.
\end{multline*}
Hint. Use the integrating factor.
11
Final Exam / FE-P7
« on: April 11, 2018, 02:47:10 PM »
Solve using (partial) Fourier transform with respect to $y$
\begin{align}
&\Delta u:=u_{xx}+u_{yy}=0, &&x>0,\label{7-1}\\
&u_x|_{x=0}= h(y),\label{7-2}\\
&u\to 0 &&\text{as }x\to +\infty\label{7-3}
\end{align}
with $h(y)=\frac{4y}{(y^2+1)^2}$.
Hint. Fourier transform of $g(y)=\frac{2}{y^2+1}$ is $\hat{g}=e^{-|\eta|}$ and $h(y)=-g'(y)$.
\begin{align}
&\Delta u:=u_{xx}+u_{yy}=0, &&x>0,\label{7-1}\\
&u_x|_{x=0}= h(y),\label{7-2}\\
&u\to 0 &&\text{as }x\to +\infty\label{7-3}
\end{align}
with $h(y)=\frac{4y}{(y^2+1)^2}$.
Hint. Fourier transform of $g(y)=\frac{2}{y^2+1}$ is $\hat{g}=e^{-|\eta|}$ and $h(y)=-g'(y)$.
12
Final Exam / FE-P6
« on: April 11, 2018, 02:46:28 PM »
Solve as $t>0$
\begin{align}
&u_{tt}-\Delta u =0, \label{6-1}\\
&u(x,y,z,0)=0,
&&u_t(x,y,z,0)=
\left\{\begin{aligned} &r^{-1}\sin(r) &&r:=\sqrt{x^2+y^2+z^2}<\pi,\\
&0 &&r\ge \pi,\end{aligned}\right.\qquad \label{6-2}
\end{align}
and solve by a separation of variables.
Hint. Use spherical coordinates, observe that solution must be spherically symmetric: $u=u(r,t)$ (explain why).
Also, use equality
\begin{equation}
r u_{rr}+2 u_r= (r u)_{rr}.
\label{6-3}
\end{equation}
\begin{align}
&u_{tt}-\Delta u =0, \label{6-1}\\
&u(x,y,z,0)=0,
&&u_t(x,y,z,0)=
\left\{\begin{aligned} &r^{-1}\sin(r) &&r:=\sqrt{x^2+y^2+z^2}<\pi,\\
&0 &&r\ge \pi,\end{aligned}\right.\qquad \label{6-2}
\end{align}
and solve by a separation of variables.
Hint. Use spherical coordinates, observe that solution must be spherically symmetric: $u=u(r,t)$ (explain why).
Also, use equality
\begin{equation}
r u_{rr}+2 u_r= (r u)_{rr}.
\label{6-3}
\end{equation}
13
Final Exam / FE-P5
« on: April 11, 2018, 02:43:36 PM »
Consider Laplace equation in the half-strip
\begin{align}
&u_{xx} +u_{yy}=0 \qquad y>0, \ 0 < x< \frac{\pi}{2} \label{5-1}\\
&u_x (0,y)=u_x(\frac{\pi}{2}, y)=0,\label{5-2}\\
&(u_y-u)(x,0)=g(x)\label{5-3}
\end{align}
with $g(x)=1$ and condition $\max |u|<\infty$.
a Write the associated eigenvalue problem.
b Find all eigenvalues and corresponding eigenfunctions.
c Write the solution in the form of a series expansion.
\begin{align}
&u_{xx} +u_{yy}=0 \qquad y>0, \ 0 < x< \frac{\pi}{2} \label{5-1}\\
&u_x (0,y)=u_x(\frac{\pi}{2}, y)=0,\label{5-2}\\
&(u_y-u)(x,0)=g(x)\label{5-3}
\end{align}
with $g(x)=1$ and condition $\max |u|<\infty$.
a Write the associated eigenvalue problem.
b Find all eigenvalues and corresponding eigenfunctions.
c Write the solution in the form of a series expansion.
14
Final Exam / FE-P4
« on: April 11, 2018, 02:41:32 PM »
Consider the Laplace equation in the sector
\begin{align}
&u_{xx} + u_{yy} =0\qquad &&\text{in } \frac{1}{4} \le x^2+y^2 < 4, y>0 ,
\label{4-1} \\
& u =1\qquad &&\text{for } x^2+y^2=4,\label{4-2}\\
& u =-1\qquad &&\text{for } x^2+y^2=\frac{1}{4},\label{4-3}\\
& u=0 &&\text{for\ \ } y=0 ,\label{4-4}
\end{align}
where $\theta$ is a polar angle.
a Look for solutions $u$ in the form of $u(r,\theta)= R(r) P(\theta)$ (in polar coordinates) and derive a set of
ordinary differential equations for $R$ and $P$. Write the correct boundary conditions for $P$.
b Solve the eigenvalue problem for $P$ and find all eigenvalues.
c Solve the differential equation for $R$.
d Find the solution $u$ of (\ref{4-1})--(\ref{4-4}).
\begin{align}
&u_{xx} + u_{yy} =0\qquad &&\text{in } \frac{1}{4} \le x^2+y^2 < 4, y>0 ,
\label{4-1} \\
& u =1\qquad &&\text{for } x^2+y^2=4,\label{4-2}\\
& u =-1\qquad &&\text{for } x^2+y^2=\frac{1}{4},\label{4-3}\\
& u=0 &&\text{for\ \ } y=0 ,\label{4-4}
\end{align}
where $\theta$ is a polar angle.
a Look for solutions $u$ in the form of $u(r,\theta)= R(r) P(\theta)$ (in polar coordinates) and derive a set of
ordinary differential equations for $R$ and $P$. Write the correct boundary conditions for $P$.
b Solve the eigenvalue problem for $P$ and find all eigenvalues.
c Solve the differential equation for $R$.
d Find the solution $u$ of (\ref{4-1})--(\ref{4-4}).
15
Final Exam / FE-P3
« on: April 11, 2018, 02:36:13 PM »
Solve by the method of separation of variables
\begin{align}
& u_{tt}- u_{xx}+ 4u =0,\qquad 0<x<\pi , \; t>0,\label{3-1}\\[2pt]
& u (0,t)= u (\pi ,t)=0,\label{3-2}\\[2pt]
& u(x,0)=f(x),\label{3-3}\\[2pt]
& u_t(x,0)=g(x)\label{3-4}\end{align}
with $f(x)=0$ and $g(x)=x^2-\pi x$. Write the answer in terms of Fourier series.
\begin{align}
& u_{tt}- u_{xx}+ 4u =0,\qquad 0<x<\pi , \; t>0,\label{3-1}\\[2pt]
& u (0,t)= u (\pi ,t)=0,\label{3-2}\\[2pt]
& u(x,0)=f(x),\label{3-3}\\[2pt]
& u_t(x,0)=g(x)\label{3-4}\end{align}
with $f(x)=0$ and $g(x)=x^2-\pi x$. Write the answer in terms of Fourier series.