I sincerely wish somebody can help me to analytically solve following ODE system. $x(t)$ $y(t)$ $z(t)$ are 3 functions of $t$, and the system is derived from an improved version of multi-compartment model:
$\left\{ \begin{array}{l} x^\prime(t)=A \cdot x(t)^2+B \cdot x+C \cdot y+D \cdot z\\ y^\prime(t)=E \cdot y(t)^2+F \cdot x+G \cdot y+H \cdot z\\ z^\prime(t)=I \cdot z(t)^2+G \cdot x+K \cdot y+L \cdot z \end{array} \right.$
A,B,C..., are just any constants, so its asymmetric.
And furthermore, when the constants also become functions of t, above system becomes
$\left\{ \begin{array}{l} x^\prime(t)=A(t) \cdot x(t)^2+B(t) \cdot x(t)+C(t) \cdot y(t)+D(t) \cdot z(t)\\ y^\prime(t)=E(t) \cdot y(t)^2+F(t) \cdot x(t)+G(t) \cdot y(t)+H(t) \cdot z(t)\\ z^\prime(t)=I(t) \cdot z(t)^2+G(t) \cdot x(t)+K(t) \cdot y(t)+L(t) \cdot z(t) \end{array} \right.$
I need the close-forms of $x(t)$ $y(t)$ $z(t)$ of both two ODE systems, so that I can do my later tasks... I will appreciate you a lot if you can offer me any solutions, hopefully a general solution for any $n$.
I was hoping to solve them by learning Riccati quadratic ODEs, but due to my limited knowledge I couldn't find solutions yet. So if you can point me any good materials, I will also appreciate. : )
Thanks a lot!!!!