Question

Question: Construct a table comparing the indicated values of y(x) using Euler's method, the improved Euler's method, and the RK4 method. Compute to four rounded decimal places. First use h = 0.1 and then use h = 0.05.

$$\begin{gathered} {\mathbf{y}}^{\text{'}}\mathbf{=}\mathbf{xy}\mathbf{+}{\mathbf{y}}^2\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1} \\ \mathbf{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)} \end{gathered}$$

Short answer

So, the required value is Rk₄.

Step-by-step solution

Definition of Euler's method

The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size and the global error (error at a given time) is proportional to the step size.

Evaluation

Given initial value problem is$y^{\text{'}}=xy+y^2,y\left(1\right)=1$

So$\text{f}(\text{x},\text{y})=\text{xy}+{\text{y}}^2,{\text{x}}_0=1,{\text{y}}_0=1$

For h = 0.1 we get

x(n)

For y(n)we get

Rk₄

I got these values by running programmed in MATALAB with this problem input.