Question

solve for x:

[x * sqrt(1+x²)] + ln[x + sqrt(1+x²)] = 25

Answer 1

%%Matlab code for finding root using newton secant bisection and false
clear all
close all

%function for which root have to find
fun=@(x) x.*sqrt(1+x.^2)+log(x+sqrt(1+x.^2))-25;
fprintf('For the function f(x)=')
disp(fun)
[root]=newton_method(fun,10,1000);
fprintf('\tValue of x using Newton method is %f.\n',root)
xx=linspace(0,5);
yy=fun(xx);
plot(xx,yy)
xlabel('x')
ylabel('f(x)')
title('x vs. f(x) plot')
hold on
plot(root,fun(root),'r*')
box on; grid on
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Matlab function for Newton Method
function [root]=newton_method(fun,x0,maxit)
syms x
g1(x) =diff(fun,x);   %1st Derivative of this function
xx=x0;            %initial guess]
fprintf('\nRoot using Newton method\n')
%Loop for all intial guesses
    n=10^-15; %error limit for close itteration
    for i=1:maxit
        x2=double(xx-(fun(xx)./g1(xx))); %Newton Raphson Formula
        cc=abs(double(fun(x2)));                 %Error
        err(i)=cc;
        xx=x2;
        if cc<=n
            break
        end
        fprintf('\tAfter %d iteration root using Newton method is %f\n',i,xx)
    end
  
    root=xx;
end

%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%