Translation
\begin{align*}
&\mathbf{v}_I=\mathbf{S}\,\mathbf{v}_B\quad\Rightarrow\,
\mathbf{\dot{v}}_I=\mathbf{\dot S}\,\mathbf{v}_B+\mathbf{S}\,\mathbf{\dot{v}}_B\\
&\text{with}\quad \mathbf{\dot S}=\mathbf{S}\,\mathbf{\omega}_B^\times\\
&\Rightarrow
\end{align*}
\begin{align*}
&\mathbf{\dot{v}}_I=\mathbf{S}\,\left(\mathbf{\omega}_B\times \,\mathbf{v}_B\right)+\mathbf{S}\,\mathbf{\dot{v}}_B\\
\end{align*}
Newton equation
\begin{align*}
&m\,\mathbf{\dot{v}}_I=\mathbf{F}_I\quad \text{or}\quad,
m\,\underbrace{\mathbf{S}^T\,\mathbf{\dot{v}}_I}_{(\mathbf{\dot{v}}_I)_B}=\mathbf{S}^T\,\mathbf{F}_I=\mathbf{F}_B
\end{align*}
you obtain
\begin{align*}
&\mathbf{S}^T\,\mathbf{\dot{v}}_I=\left(\mathbf{\omega}_B\times \,\mathbf{v}_B\right)+\mathbf{\dot{v}}_B=\frac{\mathbf{F}_B}{m}\\
&\boxed{\,\mathbf{\dot{v}}_B=\frac{\mathbf{F}_B}{m}-\left(\mathbf{\omega}_B\times \,\mathbf{v}_B\right)\,}
\end{align*}
Rotation
Euler equation in B_system
\begin{align*}
&\boxed{\,I_B\,\mathbf{\dot{\omega}}_B+\mathbf{\omega}_B\,\times \left(I_B\,\mathbf{\omega}_B\right)=\mathbf\tau_B\,}
\end{align*}
- $\mathbf S~$ transformation matrix between Body and Inertial system
- Subscript $~B~$ Body system
- Subscript $~I~$ Inertial system
Notice that with those two equations of motion, you don't get the position and the orientation (angles) of the rigid body , you need additional equations
Edit
Rotation Matrix $~\mathbf{S}~$
\begin{align*}
&\mathbf{S}=\mathbf{S}_z(\psi)\,\mathbf{S}_x(\varphi)\,\mathbf{S}_y(\vartheta)\\
&\mathbf{S}=
\left[ \begin {array}{ccc} \cos \left( \psi \right) &-\sin \left(
\psi \right) &0\\ \sin \left( \psi \right) &\cos
\left( \psi \right) &0\\ 0&0&1\end {array} \right]
\,
\left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left(
\varphi \right) &-\sin \left( \varphi \right) \\ 0
&\sin \left( \varphi \right) &\cos \left( \varphi \right)
\end {array} \right]
\,
\left[ \begin {array}{ccc} \cos \left( \vartheta \right) &0&\sin
\left( \vartheta \right) \\ 0&1&0
\\ -\sin \left( \vartheta \right) &0&\cos \left(
\vartheta \right) \end {array} \right]
\\
&\text{with}\quad \mathbf{\dot S}=\mathbf{S}\,\mathbf{\omega}^\times\\
&\Rightarrow\\
&\begin{bmatrix}
\omega_x \\
\omega_y \\
\omega_z \\
\end{bmatrix}
=\underbrace{ \left[ \begin {array}{ccc} \cos \left( \vartheta \right) &0&-\cos
\left( \varphi \right) \sin \left( \vartheta \right)
\\ 0&1&\sin \left( \varphi \right)
\\ \sin \left( \vartheta \right) &0&\cos \left(
\varphi \right) \cos \left( \vartheta \right) \end {array} \right]
}_{\mathbf{J}_R}
\,\begin{bmatrix}
\dot{\varphi} \\
\dot{\vartheta} \\
\dot{\psi} \\
\end{bmatrix}\\
&\Rightarrow
\end{align*}
\begin{align*}
& \boxed{\,\begin{bmatrix}
\dot{\varphi} \\
\dot{\vartheta} \\
\dot{\psi} \\
\end{bmatrix}=\left[ \begin {array}{ccc} \cos \left( \vartheta \right) &0&\sin
\left( \vartheta \right) \\ {\frac {\sin \left(
\varphi \right) \sin \left( \vartheta \right) }{\cos \left( \varphi
\right) }}&1&-{\frac {\sin \left( \varphi \right) \cos \left(
\vartheta \right) }{\cos \left( \varphi \right) }}
\\ -{\frac {\sin \left( \vartheta \right) }{\cos
\left( \varphi \right) }}&0&{\frac {\cos \left( \vartheta \right) }
{\cos \left( \varphi \right) }}\end {array} \right]
\,
\begin{bmatrix}
\omega_x \\
\omega_y \\
\omega_z \\
\end{bmatrix}\,}\tag 3
\end{align*}
Singularity at $~\varphi=\pi/2~$
Inertial Position vector $~\mathbf R_I~$
\begin{align*}
&\mathbf{v}_I=\mathbf{S}\,\mathbf{v}_B\\&\Rightarrow
\end{align*}
\begin{align*}
& \boxed{\,\mathbf{\dot{R}}_I=\mathbf{v}_I\,}\tag 4
\end{align*}
all together you obtained 12 first order differential equations for a rigid body solution