Question

Use an appropriate iterative method to find the solution of the equation
$\cos hx=3x$
giving your answer correct to three significant figures.

Answer 1

Here we approximate the soluion by the iterative method of the function $f\left(x\right)=\cosh hx-3x$

So, the easiest way to make the approximation is :

$\left(i\right)$ Finding the sign changing interval of the function

$\left(ii\right)$ Second you 'll be starting the iteration method with taking the midpoint of the function as start and second you would choose any of the the terminal point. Then by taking a step size of $.1$ or $.01$ we increase the value of $x$.

So,

We found that,

$f\left(0\right)=+ve$ value whereas, $f\left(1\right)=-ve$ value

Hence, The sign changing interval is $(0,1)$

now, we start with

$f\left(0.5\right)=-.37<0$

and $f\left(0\right)=1>0$

Now new sign changing interval is (0,0.5)

$x=\frac{0+.5}{2}$ = $0.25$

$f\left(.25\right)=0.281>0$

now, new interval is $(0.25,0.5)$

$x=\frac{0.25+0.5}{2}=.375$

$f\left(.375\right)=-0.052$

and, $f\left(0.25\right)>0$

interval becomes $(0.375,0.5)$

$x=\frac{0.375+.25}{2}=.3125$

$f\left(.3125\right)=.1117>0$

now solve in (.3125 , .375)

$x=\frac{.3125+.375}{2}=.34375$

$f\left(.34375\right)=.0284>0$

$x$ may be equal to $.359$

Hence, The solution of this equation

$\cos hx=3x$ is $x=0.359$