# differential equations – ODE resolution system with additional parameter

I would like to solve a $$2 times 2$$ system of form
$$frac {d} {d theta} T = TA, quad T (0) = Id$$
or $$theta$$ is real and $$A$$ is of the form
$$A = begin {pmatrix} 0 & frac {e ^ {- i theta}} { lambda} \ frac {1} {36} e ^ {- i theta} left (9 lambda + 2 ( lambda-1) ^ 2 (6 cos { theta} + cos {2 theta} + 6) right) & 0 end {pmatrix},$$
with $$lambda$$ a free parameter in the unitary circle.

In particular, I'm interested in getting digital solutions to $$theta = 2 pi$$ depending on the additional parameter $$lambda$$. I'm pretty new using Mathematicaand that's what I've tried so far:

``````T[θ_] = {{T11[θ], T12[θ]}, {T21[θ], T22[θ]}};
A[θ_] = {{0, E ^ (- I θ) / λ}, {1/36 E ^ (- I θ) (9 λ + 2 (-1 + λ) ^ 2 (6 + 6 Cos[θ] + Cos[2 θ])), 0}};
sys = {T & # 39;[θ] == T[θ].A[θ]};
``````

The previous code defines the system I want to solve and I am now trying to solve it numerically. I first tried

``````NSol = NDSolve[{sys, T11 == 1, T12 == 0, T21 == 0, T22 == 1}, {T11[θ], T12[θ], T21[θ], T22[θ]}, {θ}, {θ, 0, 2 Pi}];
``````

which gives me the exit

``````NDSolve :: dupv: "Double variable θ found in NDSolve[<[<[<[<<1>>]. "
``````

I have also tried

``````Nsol2 = ParametricNDSolve[{sys, T11 == 1, T12 == 0, T21 == 0, T22 == 1}, {T11, T12, T21, T22}, {θ, 0, 2 Pi}, {λ}];
``````

which gives me exit $$T_ {11}, dots, T_ {22}$$ as ParametricFunctions dependent on each other and on $$lambda$$.

I do not know if this is the right approach and, if so, how to extract a numeric expression based on $$lambda$$ of the last output – all that I saw in the documentation are examples plotted for specific values ​​of the parameter. Any help is very appreciated.

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