Chapter 7.1, Problem 51E

To determine

To calculate:

To calculate $\frac{dy}{dx}=x-1$ and $y=2$ 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 51E

The solution is $y\approx 2.03$

Explanation of Solution

Given information:

The given function is $\frac{dy}{dx}=x-1$ and $y=2$ when $x=1$ .

It is given that $\Delta x=0.1$ , $\frac{dy}{dx}=x-1$ and the initial point is $\left(1,2\right)$ .

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

$\left(x,y\right)$	$\frac{dy}{dx}=x-1$	$\Delta x$	$\Delta y=\frac{dy}{dx}\left(\Delta x\right)$	$\left(x+\Delta x,y+\Delta y\right)$
$\left(1,2\right)$	$0$	$0.1$	$0$	$\left(1.1,2\right)$
$\left(1.1,2\right)$	$1$	$0.1$	$0.01$	$\left(1.2,2.01\right)$
$\left(1.2,2.01\right)$	$2$	$0.1$	$0.02$	$\left(1.3,2.03\right)$

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 $y\approx 2.03$