Is it possible to do this without matlab?
%%Matlab code for Euler, Improved Euler and RK4 method
clear all
close all
%Function for which solution have to do
f=@(x,y) 10./y-y./x;
hh=[0.05 0.025];
for ii=1:2
%Euler method
h=hh(ii); % amount
of intervals
x=1;
% initial x
y=1;
% initial y
x_eval=2; % at what point
we have to evaluate
n=(x_eval-x)/h; % Number of steps
x2(1)=x;
y2(1)=y;
for i=1:n
%Eular Steps
m=double(f(x,y));
x=x+h;
y=y+h*m;
x2(i+1)=x;
y2(i+1)=y;
end
yy(ii,1)=y2(end);
fprintf('\n\tThe solution using Euler Method for
h=%.3f at x(%.1f) is %f\n',h,x2(end),y2(end))
%Improved Euler method
x=1;
% initial x
y=1;
% initial y
x_eval=2; % at what point
we have to evaluate
n=(x_eval-x)/h; % step size
x3(1)=x;
y3(1)=y;
for i=1:n
%improved Euler steps
m1=double(f(x,y));
m2=double(f((x+h),(y+h*m1)));
y=y+double(h*((m1+m2)/2));
x=x+h;
y3(i+1)=y;
x3(i+1)=x;
end
yy(ii,2)=y3(end);
fprintf('\tThe solution using improved Euler
Method for h=%.3f at x(%.1f) is %f\n',h,x3(end),y3(end))
%RK4 method
x=1;
% initial x
y=1;
% initial y
x_eval=2; % at what point
we have to evaluate
n=(x_eval-x)/h; % Number of steps
x4(1)=x;
y4(1)=y;
for i=1:n
%RK4 Steps
k1=h*double(f(x,y));
k2=h*double(f((x+h/2),(y+k1/2)));
k3=h*double(f((x+h/2),(y+k2/2)));
k4=h*double(f((x+h),(y+k3)));
dx=(1/6)*(k1+2*k2+2*k3+k4);
x=x+h;
y=y+dx;
x4(i+1)=x;
y4(i+1)=y;
end
yy(ii,3)=y4(end);
fprintf('\tThe solution using Runge Kutta 4 for
h=%.3f at x(%.1f) is %f\n',h,x4(end),y4(end))
end
%exact solution
syms y(x)
eqn = diff(y,x) == 10/y-y/x;
cond = y(1) == 1;
y_ext(x)=dsolve(eqn,cond);
fprintf('\n\tExact solution for y(2) at x=2 is %f.\n',y_ext(2))
%Table for absolute error
fprintf('\n\n\t Absolute error of Approximations to y(2)\n')
fprintf('\tMethod
\th=0.005\th=0.025\tRatio\n\n')
v1=abs(y_ext(2)-yy(1,1)); v2=abs(y_ext(2)-yy(2,1));
fprintf('\tEuler
\t%e\t%e\t%f\n',v1,v2,v1/v2)
v1=abs(y_ext(2)-yy(1,2)); v2=abs(y_ext(2)-yy(2,2));
fprintf('\tImproved Euler\t%e\t%e\t%f\n',v1,v2,v1/v2)
v1=abs(y_ext(2)-yy(1,3)); v2=abs(y_ext(2)-yy(2,3));
fprintf('\tRunge kutta \t%e\t%e\t%f\n',v1,v2,v1/v2)
%%Plotting solution using Euler method
figure(1)
hold on
plot(x2,y2,'Linewidth',2)
plot(x3,y3,'Linewidth',2)
plot(x4,y4,'Linewidth',2)
xlabel('x')
ylabel('y(x)')
title('Solution plot y vs. x')
legend('Euler Method','Modified Euler','RK4
Method','Location','northwest')
grid on
%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%
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