I am studying linear systems of differential equations and I am very new to the subject. Now, given a linear system $\dot{x}=Ax$ for some matrix $A$ with real entries, when we want to find stability of the origin - the equilibrium point- we may use
The test that uses $\det{A}=\delta$ and $Tr(A)=\tau$ so that we check the conditions on the sign of $\delta,\tau$ or $\tau^2-4\delta$
We first find the equilibrium points of the system and then evaluate them using the Jacobian matrix $Df(x)$. Then, depending on the signs of the real part of the eigenvalues of $Df(x^*)$, we say that for example the equilibrium point $x^*$ is a saddle.
However, I am not sure about when to use the first one or the second one. Can someone clarify the uses for these tests?
Both tests are meant to be applied to a differential equation of the form $$ \dot {\mathbf x} = f(\mathbf x), $$ where $\mathbf x(t) \in \Bbb R^n$ for all $t \in \Bbb R$. The second test can be applied to any differential equation of this form as long as the function $f:\Bbb R^n \to \Bbb R^n$ is differentiable. The first test applies only to the specific case that $n = 2$ (so that $\mathbf x(t) = (x_1(t),x_2(t))$) and $$ f(\mathbf x) = A \mathbf x $$ for some ($2 \times 2$) matrix $A$.
If your problem is of the form $\dot {\mathbf x} = A \mathbf x$ and $A$ has size $2 \times 2$, then apply method 1 (since it is easier). In all other cases, method 1 does not work but method 2 does.