Question

By experimenting with the improved Euler’s method subroutine, find the maximum value over the interval [0,2]of the solution to the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{sin}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}$ Where does this maximum value occur? Give answers to two decimal places.

Short answer

The maximum value of the solution on the given conditions and on the interval is 2.359

Step-by-step solution

Find the equation of approximation value

Here$\mathrm{y}\text{'}=\sin (\mathrm{x}+\mathrm{y}),\mathrm{y}\left(0\right)=2$,

For $\text{x = 0,}{\mathrm{y}}_{\mathrm{o}}=2,\text{M = 200, h = 0.1,}\text{interval =.}\left[0,2\right]$

$$\begin{gathered} \mathrm{F}=\mathrm{f}(\mathrm{x},\mathrm{y})=\sin (\mathrm{x}+\mathrm{y}) \\ \mathrm{G}=\mathrm{f}(\mathrm{x}+\mathrm{h},\mathrm{y}+\mathrm{hF}) \\ =\sin (\mathrm{x}+\mathrm{y}+0.1(1+\sin (\mathrm{x}+\mathrm{y}\left)\right) \end{gathered}$$

Solve for x and y

Apply initial points $\mathrm{x}=0,\mathrm{y}=2,\mathrm{h}=0.1$

$$\begin{gathered} \mathrm{F}(0.2)=0.909297 \\ \mathrm{G}(0.2)=0.813201 \end{gathered}$$

$$\begin{gathered} \mathrm{x}=(\mathrm{x}+\mathrm{h}) \\ \mathrm{y}=\mathrm{x}+\frac{\mathrm{h}}{2}(\mathrm{F}+\mathrm{G}) \end{gathered}$$

$$\begin{gathered} \mathrm{x}=0.1 \\ \mathrm{y}=2.08615 \end{gathered}$$

Determine the all-other values

Apply the same procedure for all other values and the values are

(x = 0.77, y = 2.359)

(x = 0.78, y = 2.360)

(x = 0.79, y = 2.360)

(x = 0.79, y = 2.359)

Hence, the maximum value of the solution on the given conditions and on the interval is 2.359.