Define the matrix $A(t)=\begin{pmatrix} - p(t) & 1 \\ -1 & p(t) \end{pmatrix}$ where $t$ is a real variable and $p$ a function of $t$.
Looking for solutions $u=(u_1(t),u_2(t))$ to the equation $$ \frac{d}{dt} u = A(t) u $$ we need to use time ordered exponential, because $[A(t_1),A(t_2)]\neq 0$; since $A$ is written in terms of generators of $\text{SL}(2)$ algebra, I would expect that in this case the solution can still have a simple explicit form, but cannot work it out. Is there any hint or reference that I could use?
You can eliminate one of the variables, "reducing" to a second-order equation
$$ \dfrac{d^2}{dt^2} u_1(t) +(p'(t)-p(t)^2+1)u_1(t) = 0 $$
Since $p'(t) - p(t)^2 + 1$ is essentially arbitrary, I don't think there is a simple explicit form in general. Even in a case as simple as $p(t) = 1+t^3$, Maple doesn't find a closed form solution.