Undergrad Constant solution and uniqueness of separable differential eq

[mr.tea]

Hi,
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.

 

[haruspex]

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

Are you sure it says only? Clearly any root of g is a constant solution.

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

Suppose x > k at some t. hg might be negative here, so as t increases x decreases. But as x approaches k, g(x) approaches zero, so the slope will level off. It can never reach k unless h goes to infinity.

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

x identically zero is a solution to the differential equation without the initial value. From the foregoing, any other solution cannot cross or meet the line x=0. In particular, a solution such that x(0)=1 cannot anywhere go to x=0.