Q 3.6-9E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Use the improved Euler’s method subroutine with step size h = 0.2 to approximate the solution to  at the points x = 0, 0.2, 0.4, …., 2.0. Use your answers to make a rough sketch of the solution on [0, 2].

 

Step-by-Step Solution

Answer

xn	yn
0.2	0.617843
0.4	1.238642
0.6	1.736531
0.8	1.981106
1.0	1.997052
1.2	1.884609
1.4	1.724472
1.6	1.561836
1.8	1.417318
2.0	1.297794

Step 1: Find the equation of approximation value

Here , f$\mathrm{y}\text{'}=\mathrm{x}+3 \cos \left(\mathrm{xy}\right),\mathrm{y}\left(0\right)=0$or $0⩽x⩽2$

 

For h=0.2, x=0, y=0, N=10

 $$\begin{gathered} \mathrm{F}=\mathrm{f}(\mathrm{x},\mathrm{y}) \\ =\mathrm{x}+3 \cos \left(\mathrm{xy}\right) \\ \mathrm{G}=\mathrm{f}(\mathrm{x}+\mathrm{h},\mathrm{y}+\mathrm{hF}) \\ =\mathrm{x}+0.2+3 \cos \left(\right(\mathrm{x}+0.2\left)\right)(\mathrm{y}+0.2(\mathrm{x}+3 \cos \left(\mathrm{xy}\right)\left)\right) \end{gathered}$$

Step 2: Solve for x 1 and y 1

Apply initial points ${\mathrm{x}}_{\mathrm{o}}=0,{\mathrm{y}}_{\mathrm{o}}=0,\mathrm{h}=0.2$

$$\begin{gathered} \mathrm{F}(0,0)=3 \\ \mathrm{G}(0,0)=3.178426 \end{gathered}$$

$$\begin{gathered} {\mathrm{x}}_{\mathrm{n}+1}=({\mathrm{x}}_{\mathrm{n}}+\mathrm{h}) \\ {\mathrm{y}}_{\mathrm{n}+1}={\mathrm{x}}_{\mathrm{n}}+\frac{\mathrm{h}}{2}(\mathrm{F}+\mathrm{G}) \\ {\mathrm{x}}_1=0+0.2 \\ =0.2 \\ {\mathrm{y}}_1=0.617843 \end{gathered}$$

 

Step 3: Evaluate the value of x 2 and y 2

$$\begin{gathered} \mathrm{F}(0.2,0.617843) =3.177125 \\ \mathrm{G}(0.2,0.617843)=3.030865 \\ {\mathrm{x}}_2=0.2+0.2 \\ =0.4 \\ {\mathrm{y}}_2=1.238642 \end{gathered}$$

Step 4: Determine the all other values

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

$$\begin{gathered} \left(\mathbf{x}=\mathbf{0}.\mathbf{6},\mathbf{y}=\mathbf{1}.\mathbf{736531}\right) \\ \left(\mathbf{x}=\mathbf{0}.\mathbf{8},\mathbf{y}=\mathbf{1}.\mathbf{981106}\right) \\ \left(\mathbf{x}=\mathbf{1},\mathbf{y}=\mathbf{1}.\mathbf{997052}\right) \\ \left(\mathbf{x}=\mathbf{1}.\mathbf{2},\mathbf{y}=\mathbf{1}.\mathbf{884609}\right) \\ \left(\mathbf{x}=\mathbf{1}.\mathbf{4},\mathbf{y}=\mathbf{1}.\mathbf{724472}\right) \\ \left(\mathbf{x}=\mathbf{1}.\mathbf{6},\mathbf{y}=\mathbf{1}.\mathbf{561836}\right) \\ \left(\mathbf{x}=\mathbf{1}.\mathbf{8},\mathbf{y}=\mathbf{1}.\mathbf{417318}\right) \\ \left(\mathbf{x}=\mathbf{2},\mathbf{y}=\mathbf{1}.\mathbf{297794}\right) \end{gathered}$$

Step 5: Plot a graph

Hence the solution is 

xn	yn
0.2	0.617843
0.4	1.238642
0.6	1.736531
0.8	1.981106
1.0	1.997052
1.2	1.884609
1.4	1.724472
1.6	1.561836
1.8	1.417318
2.0	1.297794