MA 221: The SandBox


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7 thoughts on “MA 221: The SandBox

  1. Hey Prof. Nevo, I have a question regarding a particular Laplace transform,

    L(ft) = tsint

    Can I have a hint as to how to compute this transform using the definition of the Laplace transform? I’ve tried tabular integration by parts, but not surprisingly the function repeats indefinitely and I cannot get it to resemble the original function upon integrating it in this way. Any help you can offer is greatly appreciated.

  2. Aori,

    Can you tell me how to factor the auxiliary equation that corresponds to the differential equation \(2y^{\prime \prime \prime} +3y^{\prime \prime}+y’-4y =0 \). I tried to set it equal to 0 but I don’t know how to factor this.

    1. The auxiliary equation that corresponds to the given differential equation is
      $$
      2r^3 + 3r^2 + r -4 =0.
      $$
      Unfortunately the zeros of this equation are complicated; there is only one real zero:
      $$
      \frac{1}{6} \left(-3+\sqrt[3]{216-3 \sqrt{5181}}+\sqrt[3]{3
      \left(72+\sqrt{5181}\right)}\right).
      $$

      The other zeros are imaginary:
      $$
      -\frac{1}{2}-\frac{1}{12} \left(1 + i \sqrt{3}\right) \sqrt[3]{216-3
      \sqrt{5181}}-\frac{\left(1 – i \sqrt{3}\right) \sqrt[3]{72+\sqrt{5181}}}{4\
      3^{2/3}}
      $$
      and
      $$
      -\frac{1}{2}-\frac{1}{12} \left(1 – i \sqrt{3}\right) \sqrt[3]{216-3
      \sqrt{5181}}-\frac{\left(1 + i \sqrt{3}\right) \sqrt[3]{72+\sqrt{5181}}}{4\
      3^{2/3}}.
      $$

  3. Aori,
    $$
    y = Ce^3x +1
    $$
    is a solution to
    $$
    \frac{dy}{dx}= − 3y −3.
    $$
    Graph several of the solution curves using the same coordinate axes. How do I graph this? Can u show me like a visual?

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