For the initial-value problem dy/dt +y^2 = t , y(0) = 1 that we have just considered, apply Euler’s method to estimate the value of y(1/2) using h = 0.1.

Question

For the initial-value problem

[latex] \frac {dy}{dt}+y^{2} = t , y(0) = 1[/latex]

that we have just considered, apply Euler’s method to estimate the value of
y(1/2) using h = 0.1.

Accepted Answer

At the end of this section, the implementation of Euler’s method in a spreadsheet such as Excel will be discussed. Here, we simply report the results of such a computer implementation. If we use a step size of h = 0.1, we see that we will take five steps to move from [latex]t_{0}[/latex] = 0 to [latex]t_{5}[/latex] = 0.5, the point at which we seek to approximate y. Doing so yields the output shown in table 2.1.

Table 2.1:Euler’s method applied to the IVP [latex]y´= t −y^{2}, y(0) = 1[/latex], using h = 0.1

[latex]y_{n}[/latex]	[latex]t_{n}[/latex]
1	0
0.9	0.1
0.829	0.2
0.7802759	0.3
0.749392852	0.4
0.733233887	0.5

With just five steps, we can see in the direction field in figure 2.8, together with a piecewise linear plot of the approximate solution, that we have an apparently good estimate in the above table for how the actual solution to this IVP behaves on this interval.

Question 2.6.1: For the initial-value problem dy/dt +y^2 = t , y(0) = 1 that...

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