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:

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