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september 26

  • A brine tank of volume \(V\) get an influx of \(p\%\) brine at the rate \(r\), which mixes instantaneously and flows away at the same rate \(r\). If the tank was filled with fresh water initially, when it will have
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old (mock) midterm

  1. Consider the force function
    \[
    f(y)=\sin(y)+1/2.
    \]
    1. Draw the graph of the potential function
      \[
      U(y)=-\int_0^y f(\eta)d\eta.
      \]
    2. Sketch the level sets of \( v^2/2+U(y)=C\) for \(C=-1,0,1\) on \((y,v)\) plane.
    3. Sketch the solutions of
      \[
      y”=\sin(y)+1/2,
      \]
      passing through \((0,0)\).
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september 25

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

Lyapunov’s direct method, cont’d

Convenient Lyapunov functions: homogeneous ones.

Strict homogeneous Lyapunov function for a linear system remains one after a perturbation of the system.

Examples.

Good homogeneous functions: quadratic forms.

Quadratic forms

Coordinate forms. …

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math285: the week of september 22

There will be no new homework: review the material covered thus far for the upcoming midterm (on 9.29). The midterm problems will be similar to those addressed in homework and in the class.…

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september 19

  1. Sketch the vector fields for each value of the parameter; indicate equilibria and their stability:
    • \[
      x’=3\sin(x)+x+a, a=-3,0,3;
      \]
    • \[
      x’=\tan(x)-3x+a, a=-5,0,5;
      \]
    • \[
      x’=e^x-x+a, a=-1,0,1;
      \]
  2. Find the three consecutive Picard iterations for the equation.
    Solve the differential equation;
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ece515: homework 2

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of september 25.

  1. Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\), and the operator \(D:V\to V\) of differentiation. Compute
    \[
    \exp(tD).
    \]
  2. Let
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september 18

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

Remarks
  • What operators can appear as state transition (fundamental) ones, \(\Phi(t)\)?
  • For LTVs, any operator \(\Phi\) with \(\det(\Phi)>0\) can be a fundamental solution; for LTIs the situation is far more complicated (say, the

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september 16

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

linear time dependent systems

Generalizing the notion of state transition matrix, we can address also the time-dependent systems
\[
\dot{x}(t)=A(t)x(t).
\]
Here, we dove the equation on \([t_1,t_2]\) with \(x(t_1)=x\); the value of solution at …

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math285: the week of september 15

Review sections 2.2, 2.4, 2.6 of the textbook (Edwards and Penney). Listen to the video lectures.

Additional reading (highly recommended!): chapter 1 of Ordinary differential equations : a practical guide by
Bernd J. Schroers (CUP, 2011).

Homework (due by …

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september 12

  1. Solve linear ODE of 1st order:
    1. \[
      y’+y\tan{x}=\sec{x};
      \]
    2. \[
      xy+e^x=xy’;
      \]
    3. \[
      (xy’-1)\ln{x}=2y.
      \]
  2. Reduce the ODE to a linear one, and solve:
    1. \[
      (x+1)(y’+y^2)=-y;
      \]
    2. \[
      xy^2y’=^2+y^3;
      \]
    3. \[
      xyy’=y^2+x.
      \]
  3. Reduce order of the equation, and solve:
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