I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us leave apart the technical background of the curve. My doubt is regarding the mathematical part of it. More precisely how to solve it
Equation:-
$\mathbb{M}(q)*\ddot{q}+\mathbb{K}*(q-q^0)+\mathbb{G}(q)+A(q,\dot{q})=0$
Specifications : -
1)where $q=(k_0,k_1,k_2,0,0,k_{n-1})$ , $k_i$ = $f_i(t)$, {$i = 1$ to $n-1 $ }. It means k is a function of 't' (time) . $q^0=(c_0,c_1,c_2,0,0,c_{n-1})$ where $c_i$ ($i = 1$ to $n-1 $) are constants, not variables like $k_i $. $\frac{\mathrm{d} }{\mathrm{d} t}q =\dot{q} $
2) Matrix dimensions
a)$\mathbb{M}(q) = n\times n $ , $q=q^0=\ddot{q}= n\times 1$, $\mathbb{M}(q)*\ddot{q} = n\times 1$
b)$\mathbb{K} = n\times n $ , constant matrix, $\mathbb{K}*(q-q^0) = n\times 1$ (Hint if needed :: it gives stiffness related part)
c) $ \mathbb{G}(q) = n\times 1$,(Hint if needed :: it gives gravity related part)
d) $A(q,\dot{q}) = n\times 1$
Doubts:-
1) How do we find the solution for this system of differential equation? Is there any closed form exists? if not, What is the be most suitable numerical method available for it?
2) It was advised to solve it my semi implicit Euler method. How do we do that? Is there any source related to that method available?(I have only one source https://stackoverflow.com/questions/3897424/implementing-semi-implicit-backward-euler-in-a-1-dof-mass-spring-system )
3) Is there any parallel methods available for such cases (means system of equations)?
4) In numerical methods we need to give initial value of $\dot{q}$ and proceed. Can we have any other alternatives?