An ODE confusion I was thinking about a ODE problem recently when I was reading about dynamical system. In school we used to solve the ODE problem $\frac{dx}{dt}=\sqrt{1-x^2}, x=0, t=0$ as $x=\sin(t),$ which will have the graph

Now in dynamical system we can see that the fixed points are $\pm 1$ so specifically we can observe if the solution hits $1$ or $-1$ it should not increase or decrease from there. Specifically if we draw the phase diagram we can conclude that the solution passing through $(0,0)$ should look like

and it seems reasonable. So I am surprised that we were taught wrong for many days. Isn't it? Or, am I making any mistake?

Since $\frac{dx}{dt}=\sqrt{1-x^2}\geq 0$, the solution should be increasing (not necessarily strictly)!
The global solution to the Cauchy problem
$$\begin{cases}\frac{dx}{dt}=\sqrt{1-x^2}\\
x(0)=0
\end{cases}$$
is
$$x(t)=\begin{cases}
\sin(t) & t\in [-\pi/2,\pi/2],\\
1      & t\geq \pi/2,\\
-1      & t\leq -\pi/2.
\end{cases}$$
