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december 5

  1. Solve the Dirichlet problem in the disk \(D=\{x^2+y^2\leq 4\}\) (i.e. find a harmonic function \(u\) such that on the boundary \(u\) is equal to
    1. \[
      x^3;
      \]
    2. \[
      2x^2+y^2;
      \]
    3. \[
      y^3-y.
      \]
  2. Let \(u\) be a harmonic function in
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november 21

  1. Find sin transform of the function on \([0,1]\)
    given by
    1. \(1\);
    2. \(\cos(\pi x)\);
    3. \(x\).
  2. Solve the heat equation
    \[
    u_t=\frac{1}{2} u_{xx},
    \]
    on \([0,1]\) with boundary conditions \(u(0,t)=u(1,t)=0\) and initial values
    1. u(x,0)=\(1\);
    2. u(x,0)=\(\cos(\pi x)\);
    3. u(x,0)=\(x\).
  3. Solve the heat equation
    \[
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math285, week of november 17

Read the textbook, chapters 9.5 and 9.7.

Videos to watch.

Homework (due by Monday, 12.1):

  1. Find the cosine transform of \(\sin(x)\) on \([0,\pi]\).
  2. Using separation of variables, solve the heat equation on the interval \([0,\pi]\):
    \[u_{t}=u_{xx}; \quad u_x(0,t)=u_x(\pi,t)=0; u(x,0)=\sin(x)
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november 14

  1. Find the solutions of the wave equation at time \(t=1/6,1/3,1/2\)
    \[
    u_{tt}=u_{xx}, u(0,t)=u(1,t)=0,
    \]
    on \([0,1]\)
    with initial data \(u_t(x,0)=0\) and \(u(x,0)\) is as shown:

    wave_1a wave_1b wave_1c

  2. Same question, but the initial data are
    \(u(x,0)=0\) and \(u_t(x,0)\) is as shown:

    wave_1a wave_1b wave_1c

  3. Sketch the
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math 285, week of november 10

Read the textbook, chapters 9.4 and 9.6.

Videos to watch.

Homework (due by Monday, 11.17):

  1. Find sine transform of the function (\T(x)\) on \([0,1]\) equal to \(x\) for \(0\leq x\leq 1/2\) and to \(1-x\) for \(1/2\leq x \leq1\).
  2. Using
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november 6

  • Find Fourier series for \(2\pi\)-periodic function defined on \(-\pi,\pi\) by
    1. \[
      \cos(2x)\sin^2(x);
      \]
    2. \[
      \cos^3(x);
      \]
    3. \[
      f(x)=\left\{\begin{array}{rcl} 1& \mathrm{if}& -\pi\leq x<0;\\
      -1 & \mathrm{if}& 0\leq x \leq \pi.\\
      \end{array}\right.
      \]
  • Find the Fourier series for the function which is
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    october 31

  • Find Fourier series for
    1. \[
      2-|x|
      \]
    2. \[
      2|x|-1
      \]
    3. \[
      1+|x|.
      \]
  • Find the Fourier series for the function which is 1 on the interval \([a,b]\), and 0 elsewhere on \([-\pi,\pi]\), if
    1. \[
      a=-\pi, b=-\pi/3;
      \]
    2. \[
      a=-\pi/3, b=\pi/3;
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    math285, week of october 27

    Read the textbook, chapters 9.1-9.3.

    Many videos to watch.

    Homework (due by Tuesday, 11.6):

    1. Find Fourier transform of the function on \((-\pi,\pi)\) equal to \(0\) for \(-\pi<x<0\) and to \(1\) for \(0\leq x<\pi\).
    2. Find cosine expansion of \(\sin(2x)\) on
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    october 24

    1. Consider a mass \(m\) on a spring with spring constant \(k\) oscillating on a surface with the damping constant \(c=.01\).
      1. If \(m=1, k=16, c=.01\), find, approximately, how much shrinks the amplitude after each swing.
      2. Find the maximum amplitude of the
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    october 17

      1. Solve using variation of parameters:
      2. \[
        y”+3y’+2y=1/(e^x+1);
        \]
      3. \[
        y”+y=1/\sin(x);
        \]
      4. \[
        y”+4y=2\tan(x).
        \]
    1. One of the solutions of the linear differential equation of the second order
      \[
      y”+a_1(t)y’+a_2(t)y=0
      \]
      is given; find another one using the Liouville formula
      \[
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