is $ \frac{dy}{dx}=x^2y$ stable or unstable?
I know that an equilibrium solution is said to be stable if all nearby solutions converge towards the equilibrium solution.
I analytically plotted the slope field
are the nearby solutions moving away or moving towards? How can I tell, I mean it is converging if I look at the graph from right to left or for $x$ approaches negative infinity. And similarly, I can look at the graph from left to right, the solutions are diverging so it is unstable?
Is there a quick away to find the stability of equilibrium solution? Without visualising through the slope fields? Thanks.
Look from left to right.
Near the equilibrium $y = 0$ (i.e. the $x$-axis), the slopes are pointing away from it (up in the upper half plane, down in the lower half plane). That is, a solution starting near $y = 0$ will increase and therefore move away from $y =0$ if it is $> 0$, and similarly if it is $< 0$.
Therefore the zero solution is unstable.
Without the slope field, you can see from the formula that $y' > 0$ if $y > 0$ and $y' < 0$ if $y < 0$. Then the same conclusion works.