MATH 2574H Lecture Notes - Euler Method
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10:09 am: brief recap of equilibrium solutions, numerical methods for solving first order odes. Definition: equilibrium solutions (or critical points) of autonomous equations occur where f(x) = 0. Any root c of f(x) = 0 yields the constant solution x = c. We can learn a lot about hte behavior of the solutions of autonomous odes without actually solving them, but instead studying equilibrium solutions. In particular we ca nstudy the stability of each critical point. Definition: a critical point x = c is stable if for every > 0 , there is a > 0 such that |x(0) - c| < --> It is relatively simple to study the stability of the critical points (set derivative equal to 0) and find the roots. To be stable both sides need to be moving into it if semistableonly one side doesfor unstable, both sides move away from it. Let"s say we have a derivative and some initial valuecan"t solve it analytically.
