System of ODE's - Positivity on each coordinate

Question

I have a system of ODE's

$$x'(t)=f(x,y,t)\\y'(t)=g(x,y,t)$$

s.t. $f(0,y,t)\equiv 0$ and for an initial condition $x(0)=a>0$, $y(0)=b$ I have solution $x_1(t),y_1(t)$.

Could $x_1(t)$ be negative on some $t^*$?

I think no, because taking $x\equiv 0$ the system becomes

$$y'(t)=g(0,y,t)$$

Let $y_2(t)$ be the solution of this least equation s.t. $y_2(t^{**})=y_1(t^*)$ for some $t^{**}$. So, $x_2(t)\equiv 0, y_2(t)$ is a solution to the system and contains point $(0,y_1(t^*))=(x_1(t^*),y_1(t^*))$, contradiction.

Is it correct?

Thank you so much.

Answer 1

Take $f(x,y,t)=-x^{2/3}t$ and $g(x,y,t)=0$. A solution to your system is $$x(t)=\Bigg(-\frac{t^2}{6}+a^{1/3}\Bigg)^3$$ $$y(t)=b$$ As you can see $x(t)$ will eventually assume negative values.