Let's say we have an ODEs system of the type: \begin{eqnarray} \frac{\partial \vec{y}(t)}{\partial t}=f(t,\vec{y}(t),\partial_{t}\vec{y}(t) ), \end{eqnarray} with $t\in[0,\infty)$. We want now to scale the independent variable $t$ so that $t=x/1-x$ for $x\in [0,1)$. Then, upon computing the RHS of this equation, we have: \begin{eqnarray} \frac{\partial\vec{y}}{\partial x} = \left(\frac{d t}{d x}\right)\frac{\partial \vec{y}}{\partial t}. \end{eqnarray} How can we correctly account for such transformation in the ODE? What I am doing: I always evaluate $f(t,\vec{y}(t),\partial_{t}\vec{y}(t) )$ with each evaluation of the RHS; for example, suppose I want to calculate a single differential step with a Runge-Kutta of 4th order; then the RHS needs to be evaluated 5 times to get a solution point in the interval $[0,1)$. At the end of each of the 5 evaluations, since I want $\frac{\partial\vec{y}}{\partial x}$, I multiply the whole result by the prefactor $\left(\frac{d t}{dx}\right)$, is that correct? In other words, I am just multiplying the $k_{i}$ of the RK implementation by the corresponding factors given by $\left(\frac{d t}{d x}\right)$.
Explicitly, knowing the solution at $(t_{n},y_{n})$, the first RK coefficients become: \begin{eqnarray} k_{1} &=&f(t_{n},y_{n})\to k_{1}.\left(\frac{dt}{dx}\right)\bigg|_{x=x(t_{n})}\nonumber\\ k_{2} &=&f(t_{n}+h/2, y_{n}+hk_{1}/2)\to k_{2}.\left(\frac{dt}{dx}\right)\bigg|_{x=x(t_{n}+h/2)}\nonumber\\ &\vdots & \end{eqnarray} Is this the correct way to perform the scaling?