# 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.

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

R2021b

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