How to solve for P(t) relating two sets of state variables?

I have two sets of state variables, A and Ap, and I need to solve for P(t), which relates the two. The equation to do so is Ap = inv(P)*(A*P-d/dt(P)). Does anyone know how I can solve for P(t)? I am stumped because I am unsure as to how to handle the inverted matrix of P and the derivative of the P matrix.

A = [t 1; 1 t];
Ap = [0 1; 2-t^2 2*t];

Answer 1

[KSSV]

YOu do hand calculation and get p(t)..on solving the equation becomes:

p(t+1) = p(t)+Ap*(I-A)*dt ;   % I have used derivative formula for dp/dt

Using loop and initial conditions for p(1)

A = @(t) [t 1 ; 1 t] ;
Ap = @(t) [0 1 ; 2-t.^2 2*t] ;
t0 = 0. ; dt = 0.01; t1 = 10. ;  % times 
t = t0:dt:t1 ;  % time step 
% intial conditions 
p = cell(1,length(t)) ;
p{1} = rand(2) ;
% time intergaration 
for i = 2:length(t)
    p{i} = p{i-1}+Ap(t(i))*(eye(2)-A(t(i)))*dt ;
end

[Andrew Poissant]

Hello, thank you for the reply. Is there any way to get the answer in analytical form? Maybe ode45 would be used here? I need the P(t) matrix in terms of t in the end. Thanks!