-How can I sketch the phase lines for these differential equations? What does it mean and how can I identify the equilibrium points as sinks, sources, or nodes? Letís say the equations:
dy/dt = (y-1) sin y and dy/dt = 3 y^3 Ė 12y^2
I have no idea how to start this one, how can I label each equilibrium point as I sketch these two differential equation. Iím guessing but since I have dx/dt= f(x,y) and dy/dt= g(x,y). At each point, (x,y), then dy/dx= g(x,y)/f(x,y). Should I choose a number of points in the plane, and calculate dy/dx for each of them. Because that will give me the slope of the line through that point that is tangent to the solution curve through that point.
-Now, what about if a differential equation and various initial conditions are specified. How can I sketch the graphs of the solution satisfying these initial conditions? Suppose we use the equations above, to make it more simple and understandable,
dy/dt = (y-1) sin y , and letís use y(0) = 1, y(-2) = -1, y(0) = Ĺ, y(0) = 2
same for dy/dt = 3 y^3 Ė 12y^2 but letís use y(0) = 1, y(1) = 0, y(0) = -7, y(0) = 5
After computing how can I put all the graphs on one pair of axes.
This one Iím completely blank