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Q 3.6-9E

Expert-verified
Found in: Page 130

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# 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].

 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
See the step by step solution

## Step 1: Find the equation of approximation value

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

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

$\mathrm{F}=\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\phantom{\rule{0ex}{0ex}}=\mathrm{x}+3\mathrm{cos}\left(\mathrm{xy}\right)\phantom{\rule{0ex}{0ex}}\mathrm{G}=\mathrm{f}\left(\mathrm{x}+\mathrm{h},\mathrm{y}+\mathrm{hF}\right)\phantom{\rule{0ex}{0ex}}=\mathrm{x}+0.2+3\mathrm{cos}\left(\left(\mathrm{x}+0.2\right)\right)\left(\mathrm{y}+0.2\left(\mathrm{x}+3\mathrm{cos}\left(\mathrm{xy}\right)\right)\right)\phantom{\rule{0ex}{0ex}}$

## Step 2: Solve for x1 and y1

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

$\mathrm{F}\left(0,0\right)=3\phantom{\rule{0ex}{0ex}}\mathrm{G}\left(0,0\right)=3.178426\phantom{\rule{0ex}{0ex}}$

${\mathrm{x}}_{\mathrm{n}+1}=\left({\mathrm{x}}_{\mathrm{n}}+\mathrm{h}\right)\phantom{\rule{0ex}{0ex}}{\mathrm{y}}_{\mathrm{n}+1}={\mathrm{x}}_{\mathrm{n}}+\frac{\mathrm{h}}{2}\left(\mathrm{F}+\mathrm{G}\right)\phantom{\rule{0ex}{0ex}}{\mathrm{x}}_{1}=0+0.2\phantom{\rule{0ex}{0ex}}=0.2\phantom{\rule{0ex}{0ex}}{\mathrm{y}}_{1}=0.617843\phantom{\rule{0ex}{0ex}}$

## Step 3: Evaluate the value of x2 and y2

$\mathrm{F}\left(0.2,0.617843\right)=3.177125\phantom{\rule{0ex}{0ex}}\mathrm{G}\left(0.2,0.617843\right)=3.030865\phantom{\rule{0ex}{0ex}}{\mathrm{x}}_{2}=0.2+0.2\phantom{\rule{0ex}{0ex}}=0.4\phantom{\rule{0ex}{0ex}}{\mathrm{y}}_{2}=1.238642$

## Step 4: Determine the all other values

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

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

## 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