Let us try to explore the structure of an IVP graphically by considering a specific example:\[\dfrac{dy}{dt} = y - y^2, \quad \quad \quad y(0) = 0.1.\] Suppose, we are interested in the solution \(y(t)\) over the domain \(t \in [0, 10]\).
A jupyter notebook, which accompanies this blog, is available here on github.
Let us consider the 2D domain (y versus t) on which the solution to the IVP is shown as a thick blue line. This is the solution y(t), which satisfies the initial condition, and the differential equation.
We can look at any point (t,y) on this domain, and ask "what is f(y,t) here?"
Note \(dy/dt = f(y,t)\) is the "slope"; it can be positive, negative or zero. The only restriction is that it cannot point "backwards" in the direction of negative t.
To visualize the slope at each grid point, we can set the horizontal component \(u = 1\), and the vertical component \(v = f(y,t)\), and normalize by dividing each component by \(\sqrt{u^2 + v^2}\).
Therefore, the function f(y,t) completely defines the field of arrows. The streamlines, if you will.
The initial condition \((t_0, y_0)\) tells us where to start in this field; where to "drop the feather" in the river, to be guided and carried away by the streamlines.
The jupyter notebook linked to above, lets you play around with different problems, and different initial conditions. Take a spin.
A jupyter notebook, which accompanies this blog, is available here on github.
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| click to enlarge |
We can look at any point (t,y) on this domain, and ask "what is f(y,t) here?"
Note \(dy/dt = f(y,t)\) is the "slope"; it can be positive, negative or zero. The only restriction is that it cannot point "backwards" in the direction of negative t.
To visualize the slope at each grid point, we can set the horizontal component \(u = 1\), and the vertical component \(v = f(y,t)\), and normalize by dividing each component by \(\sqrt{u^2 + v^2}\).
Therefore, the function f(y,t) completely defines the field of arrows. The streamlines, if you will.
The initial condition \((t_0, y_0)\) tells us where to start in this field; where to "drop the feather" in the river, to be guided and carried away by the streamlines.
The jupyter notebook linked to above, lets you play around with different problems, and different initial conditions. Take a spin.
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