Confusion about ODE notation $x(t,t_0,x_0)$

Question

My book is denoting the solution to $$x'(t)=f(t,x(t))$$ $$x(t_0)=x_0$$ as $$x(t,t_0,x_0)$$ passing through $(t_0,x_0)$. I don't understand why there is a $t$ and a $t_0$ involved.

Answer 1

You're pretty much letting the solution depend on the initial time with the given initial value.

For example, if $f(t,x(t))=x(t)$, then solving $x'(t)=x(t)$ we get the solution $x(t)=Ce^t$ for some $C\in\mathbb{R}$. Now if we consider the initial condition $x(0)=2$, then we get the solution $x(t)=2e^t$. Generalizing the initial condition to some given value $x_0$, so $x(0)=x_0$ gives us the solution $x(t)=x_0e^t$. But we may also consider the initial time as something else than $0$, say $t_0$, and then we get $x(t_0)=x_0$, which is just a translation in time, so the solution becomes $x(t)=x_0e^{t-t_0}$, which we may now interpret as a function of $t,t_0$ and $x_0$.