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### Topics - Victor Ivrii

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46
##### 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

47
##### 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

48
##### Final Exam / FE-P4
« on: December 14, 2018, 07:55:46 AM »

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: $\sec(t)=\frac{1}{\cos(t)}$.

49
##### Final Exam / FE-P3
« on: December 14, 2018, 07:54:21 AM »

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).

50
##### Final Exam / FE-P2
« on: December 14, 2018, 07:52:36 AM »

Find the general solution by method of the undetermined coefficients:
\begin{equation*}
y'''-3y''+4y'- 2y= 20\cosh(t)+20\cos(t);
\end{equation*}
Hint: All roots are integers (or complex integers).

51
##### Final Exam / FE-P1
« on: December 14, 2018, 07:50:55 AM »

Find the general solution
\begin{equation*}
\bigl[2x\sin(y) +1\bigr]\,dx  +
\bigl[4x^2\cos(y) + 3x\cot(y)+5 \sin(2y)\bigr]\,dy=0\,.
\end{equation*}
Hint: Use the integrating factor.

52
##### Final Exam / Readme before posting
« on: December 13, 2018, 05:12:18 AM »
You may post solutions now. Only typed solutions (upload only pictures in P2). Do not post "another" solution but discuss errors in the solution already posted.

If posted solution contains critical errors and you pointed at them, you may post your solution.

53
##### Final Exam / Postings
« on: December 13, 2018, 05:08:54 AM »
You may post solutions now. Only typed solutions (upload only pictures in P5 and P6). Do not post "another" solution but discuss errors in the solution already posted.

If posted solution contains critical errors and you pointed at them, you may post your solution.

As I graded P5 I wrote about grading; if other instructors and TAs have ime and wish , they will do the same.

I am supposed to submit marks no later than Fri night but not all problems are completely graded , so there could be a delay. In this case I apologize in advance.

54
##### Term Test 2 / TT2-P2 marking clarification
« on: December 04, 2018, 10:07:12 PM »
3+2+1 marking was a misprint. It is 2+2+2. Misprint (cloned) was discovered after test,

55
##### MAT334--Announcements / Preparing for Final
« on: December 01, 2018, 09:15:31 AM »
See TT2, Sample Problems for TT2 and Sample Problems for Final Exam.

See there student's solutions and my comments

56
##### MAT244--Announcements / Preparing to Final Exam
« on: December 01, 2018, 09:12:46 AM »
Preparing to final exam:

1) See http://forum.math.toronto.edu/index.php?topic=1191.0 it contains Final Exam problems for Spring, student's solutions, and my comments

2) See TT1, TT2  for this Fall, it contains Final Exam problems for Spring, student's solutions, and my comments

3) See Q7 for this Fall, it contains Final Exam problems for Spring, student's solutions, and my comments. Note that Q7 contains types of problems which are not covered by TT2 but are in Final

57
##### Quiz-7 / Q7 TUT 5102
« on: November 30, 2018, 04:12:23 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = (2 + x)( y - x),\\ &\frac{dy}{dt} = (4 - x)( y + x). \end{aligned}\right.

Bonus: Computer generated picture

58
##### Quiz-7 / Q7 TUT 5101
« on: November 30, 2018, 04:11:40 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = x - x^2 - xy, \\ &\frac{dy}{dt} = 3y - xy - 2y^2. \end{aligned}\right.

Bonus: Computer generated picture

59
##### Quiz-7 / Q7 TUT 0801
« on: November 30, 2018, 04:10:42 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = y +x(1-x^2 - y^2),\\ &\frac{dy}{dt} = -x + y(1-x^2 - y^2) \end{aligned}\right.

Bonus: Computer generated picture

60
##### Quiz-7 / Q7 TUT 0701
« on: November 30, 2018, 04:09:43 PM »
(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d)  Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = x - y^2, \\ &\frac{dy}{dt} = y - x^2. \end{aligned}\right.

Bonus: Computer generated picture

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