Chapter 7.1, Problem 64E

To determine

To find the percentage error for the initial value problem $\frac{dy}{dx}=2x-1$ using Euler’s Method.

Answer to Problem 64E

The percentage error for the initial value problem $\frac{dy}{dx}=2x-1$ is $2.04\%$ the approximated value of $f\left(1.6\right)\approx 1.92$ .

Explanation of Solution

Given information :

The $y=f\left(x\right)$ is the solution for the initial value problem $\frac{dy}{dx}=2x-1$ . $f\left(2\right)=3$ and $\Delta x=-0.1$ is used to approximate $f\left(1.6\right)$ .

Formula used:

Using the Euler’s method to compute $f\left(1.6\right)$ generate $\left(x+\Delta x,y+\Delta y\right)$ where $\Delta y=\frac{dy}{dx}\left(\Delta x\right)$ .

To calculate the percentage error: $\text{error = }\left|\frac{\text{exact }-\text{ estimate}}{\text{exact}}\right|$ .

Calculation:

Consider, $\frac{dy}{dx}=2x-1$

It is clear that $\Delta x=-0.1$ and the initial point is $\left(2,3\right)$ .

Using the Euler’s Method,

$\left(x,y\right)$	$\frac{dy}{dx}=2x-1$	$\Delta x$	$\Delta y=\frac{dy}{dx}\left(\Delta x\right)$	$\left(x+\Delta x,y+\Delta y\right)$
$\left(2,3\right)$	$3$	$-0.1$	$-0.3$	$\left(1.9,2.7\right)$
$\left(1.9,2.7\right)$	$2.8$	$-0.1$	$-0.28$	$\left(1.8,2.42\right)$
$\left(1.8,2.42\right)$	$2.6$	$-0.1$	$-0.26$	$\left(1.7,2.16\right)$
$\left(1.7,2.16\right)$	$2.4$	$-0.1$	$-0.24$	$\left(1.6,1.92\right)$

Therefore by using the Euler’s method the value of $f\left(1.6\right)$ is approximately equal to $1.92$ .

To verify:

Using the antiderivative of the differential equation, $\frac{dy}{dx}=2x-1$

  $\begin{array}{l}\frac{dy}{dx}=2x-1\Rightarrow {\displaystyle \int \frac{dy}{dx}={\displaystyle \int 2x-1}}\\ \Rightarrow y={\displaystyle \int 2x-1}dx\\ \Rightarrow y=x^2-x+C\end{array}$

Substitute the given point $f\left(2\right)=3$ (i.e.,) $\left(2,3\right)$ ,

  $\begin{array}{l}3={\left(2\right)}^2-\left(2\right)+C\\ 3=4-2+C\\ 3-2=C\\ 1=C\end{array}$

Therefore,

  $\begin{array}{l}y=f\left(x\right)=x^2-x+1\\ f\left(1.6\right)={1.6}^2-1.6+1=2.56-1.6+1=1.96\end{array}$

Thus, $f\left(1.6\right)\approx 1.92$ .

To calculate the percentage error:

  $\begin{array}{c}\text{error = }\left|\frac{\text{exact }-\text{ estimate}}{\text{exact}}\right|\\ =\left|\frac{1.96-1.92}{1.92}\right|\\ \approx 0.0204\\ \text{error }=2.04\%\end{array}$

Hence, the percentage error for the initial value problem $\frac{dy}{dx}=2x-1$ is $2.04\%$ the approximated value of $f\left(1.6\right)\approx 1.92$ .