Activity 7.3.3.
Consider the differential equation \(\frac{dy}{dt} = 6y-y^2\text{.}\)
(a)
Sketch the slope field for this differential equation on the axes provided.
(b)
Identify any equilibrium solutions and determine whether they are stable or unstable.
(c)
What is the long-term behavior of the solution that satisfies the initial value \(y(0) = 1\text{?}\)
(d)
Using the initial value \(y(0) = 1\text{,}\) use Euler’s method with \(\Delta t = 0.2\) to approximate the solution at \(t_i = 0.2, 0.4, 0.6, 0.8\text{,}\) and \(1.0\text{.}\) Record your results in the table and sketch the corresponding points \((t_i, y_i)\) on the axes provided. Note the different horizontal scale on the axes here compared to the axes above.
| \(t_i\) | \(y_i\) | \(dy/dt\) | \(\Delta y\) |
| \(0.0\) | \(1.0000\) | ||
| \(0.2\) | |||
| \(0.4\) | |||
| \(0.6\) | |||
| \(0.8\) | |||
| \(1.0\) |
(e)
What happens if we apply Euler’s method to approximate the solution with \(y(0) = 6\text{?}\)
