Question

Use the improved Euler’s method subroutine with step size h = 0.1 to approximate the solution to $\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{4}\mathbf{cos}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$, at the points $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{,} \mathbf{0}\mathbf{.}\mathbf{1}\mathbf{,} \mathbf{0}\mathbf{.}\mathbf{2}\mathbf{,}...\mathbf{,} \mathbf{1}\mathbf{.}\mathbf{0}$. Use your answers to make a rough sketch of the solution on [0,1].

Short answer

The required result is

$$\begin{gathered} \left(x=0,y=1\right), \\ \left(x=0.1,y=0.158447\right), \\ \left(x=0.2,y=0.4988\right), \\ \left(x=0.3,y=0.7418\right), \end{gathered}$$

$$\begin{gathered} \left(x=0.4,y=0.88777\right), \\ \left(x=0.5,y=0.95787\right), \\ \left(x=0.6,y=0.9730\right), \\ \left(x=0.7,y=0.952329\right) \end{gathered}$$

$$\begin{gathered} \left(x=0.8, y=0.906359\right), \\ \left(x=0.9, y=0.843228\right), \\ \left(x=1, y=0.768434\right) \end{gathered}$$

Step-by-step solution

Important formula.

For finding the values ofanduse Euler’s formula,

$$\begin{gathered} x_{n+1}=\left(x_n+h\right) \\ {y}_{n+1}=x_n+\frac{h}{2}\left(F+G\right) \end{gathered}$$

find the equation of approximation value

Here $y\text{'}=4\cos \left(x+y\right),y\left(0\right)=0$, for $0⩽x⩽1$.

For$h=0.1,x=0,y=1,N=10$

$$\begin{gathered} F=f \left(x, y\right) \\ =4 \cos \left(x+y\right) \\ G=f \left(x+h, y+h F\right) \\ =4 \cos \left(x+y+0.1+0.4 \cos \left(x+y\right)\right) \end{gathered}$$

solve for x1 and y1

Apply initial points $x_o=0,y_o=1,h=0.1$

$$\begin{gathered} F\left(0,1\right)=2.16121 \\ G\left(0,1\right)=1.00773 \end{gathered}$$

$$\begin{gathered} x_{n+1}=\left(x_n+h\right) \\ {y}_{n+1}=x_n+\frac{h}{2}\left(F+G\right) \end{gathered}$$

$$\begin{gathered} x_1=0+0.1 \\ =0.1 \\ {y}_1=0+\frac{0.1}{2}\left(2.16121+1.00773\right) \\ =0.158447 \end{gathered}$$

evaluate the value of x2 and u=y2.

Now, the value of F and G

$$\begin{gathered} F\left(0.1,0.158447\right)=3.86715 \\ G\left(0.1,0.158447\right)=2.93991 \\ {x}_2=0.1+0.1 \\ =0.2 \\ {y}_2=0.4988 \end{gathered}$$

determine the all-other values

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

$$\begin{gathered} \left(x=0.2,y=0.4988\right) \\ \left(x=0.3,y=0.7418\right) \\ \left(x=0.4,y=0.88777\right) \\ \left(x=0.5,y=0.95787\right) \\ \left(x=0.6,y=0.9730\right) \\ \left(x=0.7,y=0.952329\right) \\ \left(x=0.8,y=0.906359\right) \\ \left(x=0.9,y=0.843228\right) \\ \left(x=1,y=0.768434\right) \end{gathered}$$

plot a graph for all values.

By putting the values of x and y can draw a graph.

Therefore, this is the required result.