Question 2.6.2: Solve the IVP y= y −t , y(0) = 0.5 exactly, and use Euler’s ...

We first observe that y´= y −t is a linear first-order DE. Applying our work from section 2.3, we can determine that the solution to this equation is y =1+t +Ce^{t} . The initial condition y(0)=0.5 then implies that C =−1/2, so that the solution to the IVP is

y(t ) = 1+t − \frac {e^{t}}{2}

If we apply Euler’s method with h = 0.2 and take 5 steps to determine y_{n} at each, and also evaluate y(y_{n} ) at each stage, the resulting output is shown in table 2.2.

Table 2.2 Euler’s method applied to the IVP y´=y −t, y(0)=0.5, using h=0.2

Here, we observe the obvious pattern that the further we step away from the initial condition, the greater the error we encounter. This is a natural consequence of the use of linear approximations.

To get a further sense of how the error at a given step depends on step size, we now apply the same method with h = 0.1. Doing so produces the results in table 2.3. For ease of display and comparison to the case where h = 0.2, we only report the results from every other step.

Table 2.3
Euler’s method applied to the IVP y´= y −t, y(0) = 0.5, using h = 0.1