I am trying to fit a function of the form $X(t)=X_0 e^{\lambda t}$ to the following data on time $t$ and $X(t)$ and estimate the parameter value of $\lambda$.
$t=[0, 1 ,2 ,3,4,5,6]$ and $X(t)=[394000,500000,400000,300000,300000,290000,300000]$.
However, I want to check the fit if the exponential decay start from $t=1$. Still, $X_0$ should be the value at time $0$. Basically the decay start after a delay of a 1 time point.
How should I change the function of $X(t)$ so that the delay will start from time $1$ but time $0$ value is taken as the initial point?
How can I express this as a differential equation?