- #1

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I am learning ODE and I have some problems that confuse me.

In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))##

then ##x=k## is the only constant solution iff ##x## is a root of ##g##.

Moreover, it says "

*all other non-constant solutions are separated by the straight line x=k*".

First, why do we do this separation between finding constant and non-constant solutions?

Second, I don't understand the quoted sentence. why is that?

Third, there is an example of finding a solution to the initial value problem ##x'=2tx^3## and ##x(0)=1##. They say that the only constant solution is ##x \equiv 0##, and

"

*Therefore if ##x(t)## is a solution such that ##x(0)=1##, then, by uniqueness, ##x(t)## cannot assume the value 0 anywhere. Since ##x(0) =1 >0##, we infer that the solution is always positive*."

But how can ##x \equiv 0## be a constant solution if the solution should satisfy ##x(0)=1##, and how they got that the solution should be positive?

I am really confused and need some help with this.

Thank you.