MATLAB: Runge kutta vs ode

MATLABode

Hello,I need to tell what is the difference between ode23 and runge kutta using this equation:sin(x) + cos(y).I have written this code:
h=0.1;
xfinal=55;
x(1)=0;
y(1)=5;
f=@(x,y) sin(x) + cos(y);
for i= 1:ceil(xfinal/h)
    x(i+1)=x(i)+h;
    k1=f(x(i) , y(i) );
    k2=f(x(i) + 0.5*h,y(i)+0.5*k1*h);
    k3=f(x(i) + 0.5*h,y(i)+0.5*k2*h);
    k4=f(x(i)+ h, y(i) + k3*h );
    y(i+1)=y(i) + h/6*(k1+ 2*k2 + 2*k3 +k4);
end
plot(x,y);
hold on
[t,x] = ode23(f,[0 55] , 1)
plot(t,x)
And I got this plot :
I only spot a difference at the start,is this plot correct or did I just messed it up?

Best Answer

You are very close. In your Runge-Kuta solution, you specified Y0 = 5,
Y(1)=5;
but in your ode23 solution, you specified Y0 = 1.
[t,x] = ode23(f, [0,55], 1)
If you set these to the same value, you should get the same result for both methods. E.g, try
[t,x] = ode23(f, [0,55], 5);
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