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 Monday, 9.24):

  1. Sketch the vector fields for each value of the parameter; indicate equilibria and their stability:

    \[
    \dot{y}=y^3-3y+a, a=-3,0,3;
    \]

  2. Find the three consecutive Picard iterations for
    \[
    \dot{y}=y^2, y(0)=1.
    \]
    Solve the differential equation; compare the Taylor expansion of the result with the third Picard iteration.
  3. Can different solutions of the following differential equations touch at a point \((x,y)\)?

    1. \[
      y’=x+y^2;
      \]
    2. \[
      y”=x+y^2;
      \]
    3. \[
      y”’=x+y^2;
      \]

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