Ordinary Differential Equations
First-order, linear systems, and qualitative behavior in phase space.
An ODE relates a function and its derivatives. First-order:
$$\frac{dy}{dx} = f(x, y).$$Existence and uniqueness (Picard–Lindelöf) hold when $f$ is Lipschitz in $y$. Linear first-order $y' + P(x) y = Q(x)$ solved by integrating factor $e^{\int P}$.
For second-order linear with constant coefficients $ay'' + by' + cy = 0$, the characteristic polynomial $ar^2 + br + c = 0$ has roots $r_{1,2}$, giving solutions $e^{r_1 x}, e^{r_2 x}$ (real distinct), or oscillating exponentials (complex roots).
For autonomous 2D systems $\dot{\mathbf x} = A\mathbf x$, the eigenvalues of $A$ classify the origin: real same sign → node, opposite signs → saddle, complex with real part → spiral, pure imaginary → center.