# Interval of definition for an ordinary differential equation

by user707991   Last Updated October 10, 2019 03:20 AM - source

problem:Show the initial value problem does not have a solution on any interval containing $$t_0=0$$ Initial value problem:$$x'=f(t,x),x(0)=0$$

$$f(t,x)=\begin{cases} 1 & t \geq 0 \\ -1 & t < 0 \end{cases}$$

I am so lost on how to do this right now, it is unbelievable.Let me break down my confusion.

1.The book says if the differential equation is not a function of $$t$$, then solutions are defined for all $$t$$.

2.How does $$f$$ not being defined at $$x=0$$ have anything to do with a solution not existing at $$t=0$$?