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; compare the Taylor expansion of the result with the third Picard iteration.

    • \[
      y’=y+1, y(0)=0.
      \]
    • \[
      y’=y^2+1, y(0)=0.
      \]
    • \[
      y’=y(1-y), y(0)=1/2.
      \]
  3. Solve:

    • \[
      xy”’=y”-xy”;
      \]
    • \[
      yy”+y=(y’)^2;
      \]
    • \[
      yy”’+3y’y”=0.
      \]
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