# Going back in time in an initial value problem

Consider an initial value problem (IVP) $$y'=f(t,y)$$ with the initial value given by $$y(t=0) = 0$$.

If I need to find $$y(t^*)$$, hence finding the path for $$y$$ in $$t \in [0,t^*]$$ and $$t^*<0$$; is the problem then still an initial value problem? In other words can you go back in time in IVP.

The reason I ask is that I have an algorithm where in each step I need to solve for different $$t^*$$. I intend to use solve_ivp in Python which is based on Runge-Kutta 45 method and I want to know if there are any theoretical contradictions when I apply RK45.

I know that if I will use Eulers method then there is no problem. But what about RK45. Will I get the desired result.

This is technically still an IVP if you do an appropriate change of variables. Given your time is between $$t \in [t^*, 0]$$, make a new time variable $$\tau = -t$$ so that $$\tau \in [0, -t^*]$$ and you can modify the time derivatives accordingly. This means that you should have the differential equation $$\frac{dy}{d\tau} = -f(-\tau, y)$$ with $$y(\tau = 0) = 0$$ as your new IVP. This implies that you should be able to use Runge-Kutta approaches just fine.