Calculus- Early Transcendentals, 2021a

388 Differential Equations
Figure 10.3: A slope field for y ′ = t − y 2 .
Even when a differential equation can be solved explicitly, the slope field can help in understanding
what the solutions look like with various initial conditions. Recall the logistic equation y ′ = ky(M − y):
y is a population at time t, M is a measure of how large a population the environment can support, and k
measures the reproduction rate of the population. Figure 10.4 shows a slope field for this equation that is
quite informative. It is apparent that if the initial population is smaller than M it rises to M over the long
term, while if the initial population is greater than M it decreases to M.
Figure 10.4: A slope field for y ′ = 0.2y(10 − y).
Exercises for 10.4
In the following exercises, compute the Euler approximations for the initial value problem for 0 ≤ t ≤ 1
and Δt = 0.2. If you have access to Sage, generate the slope field first and attempt to sketch the solution
curve. Then use Sage to compute better approximations with smaller values of Δt.
Exercise 10.4.1 y ′ = t/y, y(0)=1
Exercise 10.4.2 y ′ = t + y 3 ,y(0)=1