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ece515, midterm

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Midterm.
Open book. Points for each (sub)problem shown. Full credit given for 25 points.

  1. Consider the smallest linear space \(V\) of functions on the real line containing the functions
    \(te^t, te^{-t}\) and
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math285, week of october 13

Read the textbook, chapters 3.4 and 3.6. additional reading (highly recommended).

Videos to watch.

Homework (due by Tuesday, 10.21):

    Solve using variation of parameters:
  1. \[
    y”+9y=\sec(3t);
    \]
    Answer:\((1/9)\cos(3t)\ln|\cos(3t)|+(1/3)t\sin(3t)+C_1\cos(3t)+C_2\sin(3t)\).
  2. \[
    y”+2y’+y=e^{-t}/t.
    \]
    Answer: \(e^{-t}t\ln{t}+C_1e^{-t}+C_2te^{-t}\).
  3. Consider a mass \(m=1\) on
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october 10

Solve:

    • \[
      y”-2y’+y=te^t;
      \]
    • \[
      y”-2y’+y=t^2e^t;
      \]
    • \[
      y”-2y’+y=t^3e^t;
      \]
    • \[
      y”+y=t\sin(t);
      \]
    • \[
      y”+y=t^2\sin(t);
      \]
    • \[
      y”+y=t^3\sin(t);
      \]
    • \[
      t^2y”+ty’-4y=t^{-2};
      \]
    • \[
      t^2y”-2y=t^{-2};
      \]
    • \[
      t^2y”+ty’+y=10t^3.
      \]
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math285, week of october 6

Read the textbook, chapters 3.3 and 3.5. additional reading (highly recommended): Schoers, chapter 2.

Videos (a lot of them).

Homework (due by Tuesday, 10.14):

Solve

  1. \[
    y”’+27y=e^{-3t}+27;
    \]

    Answer: \(te^{-3t}/27+1+c_1e^{-3t}+e^{3t/2}(c_2\cos(3\sqrt{3}t/2)+c_3\sin(3\sqrt{3}t/2 )\).

  2. \[
    y^{(4)}+2y”+y=e^t;
    \]
    Answer:
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ece515, homework 3

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by noon of october 14.

  1. Consider the system
    \[
    \begin{array}{ccl}
    \dot{x}_1&=&x_2\\
    \dot{x}_2&=&-ax_2-a^2x_1-x_1^3\\
    \end{array}
    \]
    For which values of \(a\) the system is
    unstable? Lyapunov stable? (Locally) asymptotically stable?
  2. Let
    \[
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october 3

  • Solve:
    • \[
      y”-3y’+2y=0; y(0)=0, y'(0)=1;
      \]
    • \[
      y”+4y’+4y=0; y(0)=0, y'(0)=1;
      \]
    • \[
      y”+y=0; y(0)=0, y'(0)=1;
      \]
  • Solve:
    • \[
      y”-3y’+2y=te^t-e^{2t}; y(0)=0, y'(0)=1;
      \]
    • \[
      y”+4y’+4y=t-e^{-2t}; y(0)=0, y'(0)=1;
      \]
    • \[
      y”+y=\cos(t)+t; y(0)=0, y'(0)=1.
      \]
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october 2

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

Quadratic forms as Lyapunov functions

If the operator defines an asymptotically stable system, it is Hurwitz. For a Hurwitz operator, a quadratic form exists which is a (strict) Lyapunov function. A strict Lyapunov function implies …

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solutions

for homework 1 and homework 2.

Thanks to Cheng!…

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math285: the week of september 29

Please read the text book, Chapter 3.1-2.

The lectures are here and here.

The homework (due by Monday night):

  1. Solve
    \[
    y”+y’-2y=0.
    \]
    Answer:
    \(c_1e^t+c_2e^{-2t}\).
  2. Find the bounded solutions of
    \[
    y”=y
    \]
    such that \(y(0)=1\).

    Answer:

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september 30

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

Lyapunov’s direct method, cont’d

For complex spaces quadratic forms are not really suitable (if one wants just a real number as a result, the signatures are all \((n,n)\)

Sylvester criterion for positive definiteness.
(Bonus: Rayleigh …

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