First-order Autonomous ODE's: Qualitative Methods

June 2, 2019

Autonomous ODE’s

Dy/dt = f(y); no independent variables on the right hand side;

Seperating variables;

Getting qualitative information without solving the equation

All isoclines are horizontal because dy/dt is not changed along with t/independent variable

Also, the integral lines are exactly the same

Critical Point

Crictical point is a point where f(y_i)=0 y_0; then f(y_0) = 0; the slope on the level y=y_0 is zero/horizontal; also, y=y_0 is a solution;

If there’s another point y_1 s.t. f(y_1)=0; then there’re two horizontal lines on the direction field; they will form two barriers b/c other curves can’t cross these lines

Steps…

  1. Find the critical points
  2. Draw the graph of f(y)
  3. …

Draw the plot of dy/dt to y; analyze how dy/dt changes along with y

Logistic equation

Stable critical point; unstable critical point; semi-stable;

Maximum harvesting rate

First-order Autonomous ODE's: Qualitative Methods - June 2, 2019 - Ruizhen Mai