I am trying to convert the second order IVP $$\frac{d^2 y}{dt^2}-ty=\mu y^3 \ \ \text{with} \ \ y(2)=1, \ \ y'(2)=4$$ into an autonomous system of first order ODEs with the parameters treated as unknowns.
We let $x_1=y$, $x_2=y'$ and $x_3=t$. Hence, $$\vec{x'}=\begin{pmatrix} x'_1 \\ x'_2 \\ x'_3 \\ \end{pmatrix}=\begin{pmatrix} x_2 \\ \mu x^3_1+x_3x_1 \\ 1\\ \end{pmatrix}=f(\vec{x}),$$ with $$\vec{x(2)}=\begin{pmatrix} x_1(2) \\ x_2(2) \\ x_3 (2)\\ \end{pmatrix}=\begin{pmatrix} 1 \\ 4 \\ 2\\ \end{pmatrix}.$$ What I have done here is convert the IVP into an autonomous system of first order ODEs. But how do I treat the parameters as unknowns?
The only parameter you have used is $\mu$ so I don't see any problem at all to define a 4th variable $x_4=\mu$ and let $x_4'=0$. I don't know why you want to treat it as a variable but by the way you can proceed exactly as for the t variable/parameter. So for instance you would get $x_2'=x_4x_1^3+x_3x_1$