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**):

- 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;

\] - 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. -
Can different solutions of the following differential equations touch at a point \((x,y)\)?
- \[

y’=x+y^2;

\] - \[

y”=x+y^2;

\] - \[

y”’=x+y^2;

\]

- \[

Could you clarify question 3? I’m not really sure what to do.

Two solutions touch at a point if they have the same value (

y) and the same derivative (slope).