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 $t=0$?

Please help these are nonhomework practice problems to prepare for a test.

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