Sketch the vector field on the real line, find all the fixed points, classify their stability, and sketch the graph of x(t) vs. t for different initial conditions. Then try for a few minutes to obtain the analytical solution for x(t).
dx/dt = x-x^3
I found the fixed points to be 0, +1, and -1. I don't know what the graph of x-x^3 looks like so I tried plugging in values to the left and right of 0, +1, and -1 to figure out their stability. I found 0 to be unstable, -1 to be semi-stable, and +1 to be stable. Now I want to draw the x(t) graph but how do I know how the lines in this graph look like? I know below -1, it moves towards -1. Above -1, it moves towards +1. And above +1, it moves toward +1.
1. Are my steps right throughout this whole problem?
2. And if so, how do I find the slopes of the lines in the x(t) graph?
3. Also, what would the "analytical solution" be? I don't know what it's asking.
dx/dt = x-x^3
I found the fixed points to be 0, +1, and -1. I don't know what the graph of x-x^3 looks like so I tried plugging in values to the left and right of 0, +1, and -1 to figure out their stability. I found 0 to be unstable, -1 to be semi-stable, and +1 to be stable. Now I want to draw the x(t) graph but how do I know how the lines in this graph look like? I know below -1, it moves towards -1. Above -1, it moves towards +1. And above +1, it moves toward +1.
1. Are my steps right throughout this whole problem?
2. And if so, how do I find the slopes of the lines in the x(t) graph?
3. Also, what would the "analytical solution" be? I don't know what it's asking.