Chapter 7.1, Problem 52E

To determine

To calculate:

To calculate $\frac{dy}{dx}=y-1$ and $y=3$ when $x=1$ with the increments of $\Delta x=0.1$ to approximate the value of y when $x=1.3$ using Euler’s method.

Answer to Problem 52E

The solution is $y\approx 3.662$

Explanation of Solution

Given information:

The given function is $\frac{dy}{dx}=y-1$ and $y=3$ when $x=1$ .

It is given that $\Delta x=0.1$ , $\frac{dy}{dx}=y-1$ and the initial point is $\left(1,3\right)$ .

To solve using Euler’s method, make a table as shown

$\left(x,y\right)$	$\frac{dy}{dx}=y-1$	$\Delta x$	$\Delta y=\frac{dy}{dx}\left(\Delta x\right)$	$\left(x+\Delta x,y+\Delta y\right)$
$\left(1,3\right)$	$2$	$0.1$	$0$	$\left(1.1,3.2\right)$
$\left(1.1,3.2\right)$	$2.2$	$0.1$	$0.22$	$\left(1.2,3.42\right)$
$\left(1.2,3.42\right)$	$2.42$	$0.1$	$0.242$	$\left(1.3,3.662\right)$

After completing each row write the point $\left(x+\Delta x,y+\Delta y\right)$ that found at the point $\left(x,y\right)$ in next row. Continue to find $\left(x+\Delta x,y+\Delta y\right)$ until the x-co ordinate reaches $1.3$ .

Hence, the solution is $y\approx 3.662$