Q 3.6-14E - Fundamentals Of Differential Equations And Boundary Value Problems

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.

Step-by-Step Solution

Answer

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

Step 1: Find the equation of approximation value

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

 

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

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

 

Step 2: Solve for x and y

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

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

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

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

 

Step 3: 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.