Example 1 - Euler approximation

Compute the Euler approximation to the initial value problem

y' = t - y², y(0) = 0

for 0 ≤ t ≤ 1, with Δt = 0.2.

Notice that the differential equation is not linear because of the y², and the variables are not separable, so we cannot solve the equation by the methods of the preceding sections. Given Y(t), the next value Y(t + Δt) is computed by the rule

We record the values in a table. The third column gives the change in Y(t). The graph of Y(t), shown in Figure 14.4.2, is obtained by connecting the points (t, Y(t)) in the table by straight lines.

Figure 14.4.2 Example 1

Work the same problem with Δt = 0.1.

We now consider Euler approximations with infinitesimal Δt. These approximations cannot be computed directly but are useful in showing that a differential equation has a solution.

The Euler approximation Y(t) depends on both t and the increment size Δt. Now let Δt be positive infinitesimal. By the Transfer Principle, Y(t + Δt) is still given by the rule

Y(t + Δt)- Y(t) = f(t,Y(t))Δt.

Intuitively, the graph of Y(t) as a function of t is formed from infinitesimal line segments, and the segment from t to t + Δt has slope f(t, Y(t)), as in Figure 14.4.3. The next theorem shows that the Euler approximation for infinitesimal Δt is infinitely close to a solution of the initial value problem.