# Solving system of four second order ODEs

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Jake Barlow on 22 Jan 2022
Commented: Jake Barlow on 23 Jan 2022
Hi,
I am trying to solve a system of four second order ordinary differential equations with the initial conditions . However after running the following code
syms U1(t) V1(t) U2(t) V2(t)
%Constants
alpha = 0.5;
dU1 = diff(U1, t);
dU2 = diff(U2, t);
dV1 = diff(V1, t);
dV2 = diff(V2, t);
t0 = 0; %initial time
tf = 5; %final time
%initial conditions
c1 = 1; %U(0)
c2 = 0; %U'(0)
c3 = 1; %V(0)
c4 = 0; %V'(0)
y0 = [c1 c2 c3 c4];
eq1 = diff(U2, 2) == alpha*dV2;
eq2 = diff(V2, 2) == -alpha*dU2;
eq3 = diff(U1, 2) == alpha*dV1;
eq4 = diff(V1, 2) == -alpha*dU1;
vars = [U1(t); V1(t); U2(t); V2(t)];
V = odeToVectorField([eq1,eq2,eq3,eq4]);
M = matlabFunction(V, 'vars', {'t','Y'});
interval = [t0 tf]; %time interval
ySol = ode45(M,interval,y0);
tValues = linspace(interval(1),interval(2),1000);
yValues1 = deval(ySol,tValues,1); %U(t)
yValues2 = deval(ySol,tValues,3); %V(t)
plot(yValues1, yValues2)
I get this error
Index exceeds the number of array elements. Index must not exceed 4.
Error in symengine>@(t,Y)[Y(2);Y(4).*(-1.0./2.0);Y(4);Y(2)./2.0;Y(6);Y(8)./2.0;Y(8);Y(6).*(-1.0./2.0)]
Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 106)
odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);
Could you please help? If there is another way to solve this sytem then it will be much appreciated if you can share it. Thank you.
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### Accepted Answer

Torsten on 22 Jan 2022
4 second-order ODEs need 8 initial conditions, not 4.
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Jake Barlow on 23 Jan 2022
Hi Torsten, thank you very much. I specified the initial conditions in the correct order.

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