# How can I call the highest order solution (e.g. x'') of an ode in another ode?

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Leonie Schickmann on 24 Feb 2021
Commented: Leonie Schickmann on 24 Feb 2021
I want to solve the following differential equations, a-f are known variables which are manually set before solving the equations:
u'' = -a/d*u' - b/d*u - x''
x''= (c+d)^(-1)*( - e*x' - f*x - d*u'')
I do not know how to call x'' in the first and u'' in the second ode. My approach (given below) does run, but the results are wrong (u and u' are always 0). How do I implement my equations correctly?
The code I tried to use:
tspan = [0 200];
z0 = [0; 0; 5; 0];
[t,z] = ode45(@(t,z)f(t,z),tspan, z0);
function dzdt = f(t,z)
u = z(1:2);
x = z(3:4);
% Definition of a-f here
dzdt = zeros(4,1);
dzdt(1) = u(2);
dzdt(2) = -a/d *u(2)-b/d *u(1)-dzdt(4);
dzdt(3) = x(2);
dzdt(4) = (c+d)^(-1)*(-e*x(2)-f*x(1)-d*dzdt(2));
end

Alan Stevens on 24 Feb 2021
Edited: Alan Stevens on 24 Feb 2021
You can rearrange the equations as follows to get a consistent set:
(you will need to use explicit multiplication where my equations have implicit multiplication. For example,
where I have fx(1) you will need f*x(1) etc.)
Leonie Schickmann on 24 Feb 2021
Thank you!