Question: I want to integrate a function numerically

Good day, 

I have this system of differential equations, which I solved numerically. 

Now I need to integrate the function y(t) numerically to get the corresponding expression for 

Z= exp(int y(t) dt ) (integrate from a given t to 0)

then I need to plot that system of differential equation vs (Z) with (t) as the parameter.

can you please help?

 eq1 := diff(x(t), t)-(1/6)*(6*x(t)^3*y(t)+(2*y(t)^2-2)*x(t)^2+3*y(t)*(z(t)-2)*x(t)-2*y(t)^2+2)*sqrt(3) = 0;

eq2 := diff(y(t), t)-(1/6)*(y(t)-1)*sqrt(3)*(y(t)+1)*(6*x(t)^2+2*y(t)*x(t)+3*z(t)-2) = 0;

eq3 := diff(z(t), t)-(1/3)*z(t)*sqrt(3)*(6*y(t)*x(t)^2+2*x(t)*y(t)^2+3*z(t)*y(t)-2*x(t)-3*y(t)) = 0;

eq4 := diff(s(t), t)-(1/3)*(y(t)-1)*sqrt(3)*(y(t)+1)*(6*x(t)^2+2*y(t)*x(t)+3*z(t)-2)*y(t) = 0;

eq5 := diff(Q(t), t)+sqrt(3)*((4/3-z(t)-2*x(t)^2)*y(t)+(2/3*(1-y(t)^2))*x(t))*Q(t)+2*((2/3)*y(t)+x(t))*P(t)/sqrt(3)-(2/3)*R(t) = 0;   
eq6 := diff(P(t), t)+(sqrt(3)*(1-z(t)-2*x(t)^2)*y(t)+2*(2-y(t)^2)*x(t)/sqrt(3))*P(t)+2*sqrt(3)*x(t)*Q(t)+(1/2)*R(t) = 0;       
eq7 := diff(R(t), t)-sqrt(3)*((3*x(t)^2+(3/2)*z(t)+4)*y(t)-(1/3*(1-3*y(t)^2))*x(t))*R(t)-z(t)*P(t) = 0;
eq8 := diff(T(t), t)-sqrt(3)*((6*x(t)^2+3*z(t)+1)*y(t)-(2*(1-y(t)^2))*x(t))*T(t)-(1-y(t)^2)*(Q(t)-(2/3)*P(t)) = 0;   
syst := {eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8};

ics := x(0) = .1, y(0) = .9, z(0) = .1, s(0) = .2, Q(0) = .1, P(0) = .9, R(0) = .1, T(0) = .2;
  

sol1 := dsolve({ics, op(syst)}, {P(t), Q(t), R(t), T(t), s(t), x(t), y(t), z(t)}, type = numeric, output = listprocedure);
 

 

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