2nd order euler method problem

my brain is boiling

How to solve initial value problem?

x x ̈ +x ̇ ^2+cost=0, x(0)=√6, x ̇ (0)=0

% euler method 2nd order 

clc;
clear;
close all;

h=0.1;
t = 0:h:10; 
n = numel(size(t));

     x = zeros(size(t));
  dxdt = zeros(size(t));
dx2dt2 = zeros(size(t));

   x(1) = sqrt(6);
dxdt(1) = 0;

plot(t,x);

for i = 1:n-1
    dx2dt2(i+1) = dx2dt2(i) + h*f2(t(i),dx2dt2(i));
    dxdt(i+1) = dxdt(i) + h*f1(t(i),dxdt(i));
end

function dx2dt2 = f2(t,x,dxdt)
 dx2dt2 = -(dxdt^2 + cos(t))/x;
end

function dxdt = f1(t,x,dx2dt2)
dxdt = sqrt(cost(t)-x*dx2dt2);
end

also i made another code to slove same problem

% euler method 2nd order 

clc;
clear;
close all;

h=0.1;
t = 0:h:10; 
n = numel(size(t));

     x = zeros(size(t));
  dxdt = zeros(size(t));
dx2dt2 = zeros(size(t));

   x(1) = sqrt(6);
dxdt(1) = 0;


f2 = @(t,x,dxdt) -(dxdt^2 + cos(t))/x;
f1 = @(t,x,dx2dt2) sqrt(cost(t)-x*dx2dt2);

for i = 1:n-1
    dx2dt2(i+1) = dx2dt2(i) + h*f2(t(i),dx2dt2(i));
    dxdt(i+1) = dxdt(i) + h*f1(t(i),dxdt(i));
end

plot(t,x);

thank you

Answer 1

[Jan]

You have to convert the 2nd order equation to a system of order 1 at first. Accumulating dx2/dt2 is not meaingful.

The 2nd derivative at the point [t, x, dxdt] is:

dx2dt2 = -(dxdt^2 + cos(t)) / x

You use this acceleration to calculate the velocity, and the velocity to get the position in each time step.