Topics

616

Easter and Semester End Challenge / Semester End Challenge 2

« on: April 04, 2013, 06:35:39 AM »

Both parts are separate but related problems

(A) Draw phase
\begin{equation}
\left\{\begin{aligned}
&\frac{dx}{dt}=-6xy,\\
&\frac{dy}{dt}=-3x^2+3y^2.
\end{aligned}\right.
\tag{a}
\end{equation}
s it integrable? Find equilibrium points and try to classify them.


(B) Draw phase
\begin{equation}
\left\{\begin{aligned}
&\frac{dx}{dt}=-\cos (4y)+\cos(2y)\cos (2\alpha x),\\
&\frac{dy}{dt}=\alpha \sin(2y)\sin (2\alpha x).
\end{aligned}\right.
\tag{b}
\end{equation}
Is it integrable? Find equilibrium points and try to classify them. What is connection to (I)? For calculations take $\alpha=1$ but for $\alpha=\sqrt{3}$ picture will be nicer (phase portraits are similar).

617

Easter and Semester End Challenge / Semester End Challenge 1

« on: April 04, 2013, 06:05:54 AM »

Draw phase portraits :

\begin{align}
&\left\{\begin{aligned}
&x'=-\sin(y),\\
&y'=   \sin (x);
\end{aligned}\right. \tag{a}\\
&\left\{\begin{aligned}
&x'=-\sin(y)-\alpha \sin (x),\\
&y'=   \sin (x)-\alpha \sin(y);
\end{aligned}\right. \tag{b}
\end{align}
where (a) was part of the Easter challenge and consider (b) for $\alpha=\pm 1$.

Observe the differences between these  phase portraits. Explain them.

618

Quiz 5 / Night Sections

« on: April 03, 2013, 08:07:49 PM »

\begin{equation*}
\left\{\begin{aligned}
&\frac{dx}{dt} = x(1.5-0.5x-y),\\
&\frac{dy}{dt} = y(2-y-1.125x).
\end{aligned}\right.
\end{equation*}


This problem can be interpreted as describing the interaction of two species with population densities $x$ and $y$.


(b)  Find the critical points.

(c)  For each critical point find the corresponding linear system.  Find the eigenvalues and eigenvectors of the linear system;   classify each critical point as to type, and determine whether  it is asymptotically stable, stable, or unstable.

(d) Sketch the trajectories in the neighborhood of each critical point.

(e) [Bonus]  Find, if possible, solution in the form $H(x,y)=C$ and sketch the phase portrait.

619

Easter and Semester End Challenge / Easter challenge

« on: March 28, 2013, 07:36:07 PM »

Draw phase portraits :

\begin{gather}
\left\{\begin{aligned}
&x'=-\sin(y),\\
&y'=   \sin (x);
\end{aligned}\right. \tag{a}\\
\left\{\begin{aligned}
&x'=-\sin(y),\\
&y'= \,2\sin (x);
\end{aligned}\right. \tag{b}
\end{gather}

Explain the difference between portraits and its reason

620

Term Test 2 / Official Solutions for TT2

« on: March 28, 2013, 07:28:37 PM »

Here are official solutions for TT2

http://www.math.toronto.edu/courses/mat244h1/20131/MAT244_2013S_TT2_Solutions.pdf

Too large to attach (graphics)

621

Quiz 4 / Q4--day section

« on: March 22, 2013, 04:18:05 AM »

Please post the problem and its solution (in contrast to a popular opinion I am not omniscient and unless someone advises me I am not sure which problem it was).

622

Quiz 4 / Quiz 4--Problem (night sections)

« on: March 20, 2013, 08:11:16 PM »

9.2 p 517, # 10(a,c)

Consider system
\begin{equation*}
\left\{\begin{aligned}
&\frac{dx}{dt}=(2+x)(y-x),\\
&\frac{dy}{dt}=y(2+x-x^2);
\end{aligned}\right.
\end{equation*}
(a - 1 points) Find all critical points (equilibrium solutions) and write  the linearization of the system at each critical point;

(b - 3 points) For the linearized systems at each critical point draw the phase portrait and identify its type (including stability, if applicable,  the orientation, etc.);

(Bonus - 1 point) Describe (by drawing) the basin of attraction for each  asymptotically stable point (if there is any) and determine for each
critical point whether the phase portrait will not change in the nonlinear  system.

623

MidTerm / Official solutions

« on: March 14, 2013, 01:33:41 PM »

Please find official solutions with helpful remarks

624

MidTerm / MT Problem 5

« on: March 06, 2013, 09:10:42 PM »

Solve the system of ordinary differential equations
\begin{equation*}
\left\{
\begin{aligned}
&x'_t=5x-3y,\\
&y'_t=6x-4y.
\end{aligned}
\right.\end{equation*}

625

MidTerm / MT Problem 4

« on: March 06, 2013, 09:09:26 PM »

Find a particular solution of equation
\begin{equation*}
y'''-2y''+4y'-8y=e^{3x}
\end{equation*}

626

MidTerm / MT Problem 3

« on: March 06, 2013, 09:08:26 PM »

Find a particular solution of equation
\begin{equation*}
t^2 y''-2t y' +2y=t^3 e^t.
\end{equation*}

[BONUS] Explain whether the method of undetermined  coefficients to find a particular solution of this equation applies.

627

MidTerm / MT Problem 2b

« on: March 06, 2013, 09:07:57 PM »

Find a particular solution of equation
\begin{equation*}
(t^2-1) y''-2ty'+2 y=1.
\end{equation*}
Hint: use variation of parameters.

628

MidTerm / MT Problem 2a

« on: March 06, 2013, 09:06:36 PM »

Find solution $y_2(t)$ of
\begin{equation*}
(t^2-1) y''-2ty'+2 y=0
\end{equation*}
where one of the solutions is $y_1(t)=t$ and solution $y_2$ is such that  $W(y_1,y_2)=-1$ at $t=0$ and $y_2(0)=1$.

629

MidTerm / MT Problem 1

« on: March 06, 2013, 09:05:02 PM »

Solve the initial value problem
\begin{equation*}
z'' -3z' + 2z = 2e^{3x} , \qquad z(0) = 1 , \qquad z'(0) = 0 .
\end{equation*}

630

Quiz 3 / Day Section Problem 2

« on: February 27, 2013, 07:46:41 PM »

Post problem and solution