How to write a differential equation with two enumeration

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I'm not sure what you mean under "write", but if you question is about defining 2D array of functions P, you can proceed like this:

N = 4
t = var('t')
P =  Matrix(N+1,N+1, lambda i,j: function(f'P_{i}_{j}')(t) if i+j<=N else 0)
print(P)

which gives

[P_0_0(t) P_0_1(t) P_0_2(t) P_0_3(t) P_0_4(t)]
[P_1_0(t) P_1_1(t) P_1_2(t) P_1_3(t)        0]
[P_2_0(t) P_2_1(t) P_2_2(t)        0        0]
[P_3_0(t) P_3_1(t)        0        0        0]
[P_4_0(t)        0        0        0        0]

Then you can work with functions P[i,j] as you like, including defining differential equations over them.