This applet plots solution curves and direction fields of first order differential equations of the form:
dy/dt = f(t,y)
The vector at a point [t,y(t)] is given by <t,f(t,y)> with the field being represented in the applet as a "direction field" of arrows. The arrow at a given point points in the direction of the vector at that point, but the length of the vector is not represented. You can start a solution curve at a given initial point by clicking the mouse at that point. You can also enter the t and y(t) coords of the initial point in the text input boxes labeled "t0=" and "y0=". Press return in one of these input boxes or click the green start button to start the curve. The curves are animated. At each step of the animation, a new segment of each curve is computed and drawn. The segment represents an approximation of the exact solution curve over a time interval of length dt. The approximation method can be either be a four stage, 8th order implicit Gauss method, explicit Runge-Kutta order 4, explicit Runge-Kutta order 2, or explicit Euler's method. The implicit Gauss method is implemented using functional iteration for simplicity. You can stop the progress of the curves by clicking the red stop button. The curves will also be stopped automatically if their coordinates become undefined or unreasonably large. You can clear any curves that have been drawn by clicking the Clear button. Curves will also be cleared if a new vector field is drawn. If the function f(t,y) has singularities, the user should make sure that the singularities are not located at any values t=tmin + dt, or an error will result. Click the button below and the applet with open in a new window.
cubic: y' = a + b*y + c*y^2 + d*y^3
From a qualitative point of view, the dynamics of a one-dimensional equation are quite simple; either an orbit eventually approaches an equilibrium point, or it goes to infinity. It is important, however, to understand the dependence of the dynamics of an equation on its parameters because the number of equilibria and their stability types may change as these parameters are varied. Despite the apparent simplicity of this example, the three most common qualitative changes around an equilibrium point - the ptichfork, saddle-node, and transcritical bifurcations - can be seen.
The following examples are from the book, Elementary Differential Equations and Boundary Value Problems, 7th edition, by W. Boyce and R. DiPrima.