Question

Construct a table comparing the indicated values of $\mathit{y}\mathbf{\left(}\mathit{x}\mathbf{\right)}$using Euler's method, the improved Euler's method, and the RK4 method. Compute to four rounded decimal places. First use $\mathit{h}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}$and then use $\mathit{h}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.

$\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{c}\mathit{o}\mathit{s}{\mathit{y}}^{\mathbf{2}\mathbf{}}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$
$\mathit{y}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{)}\mathbf{,}\mathit{y}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{,}\mathit{y}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{)}\mathbf{,}\mathit{y}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{4}\mathbf{)}\mathbf{,}\mathit{y}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{)}$

Short answer

So, the required value is $R{k}_4$.

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 $y=\sin x^2+\cos y^2,y\left(0\right)=0$

So $f(x,y)=\sin x^2+\cos y^2,x_0=0,y_0=0$

For $h=0.1$we get

$x\left(n\right)$

For $y\left(n\right)$we get

$R{k}_4$

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