First-order linear ODEs

\[\frac{dy}{dt} + p(t)y = q(t)\]

Solved by the integrating factor $\mu(t) = e^{\int p(t)\,dt}$:

\[y = \frac{1}{\mu(t)}\left[\int \mu(t)q(t)\,dt + C\right]\]

Second-order linear ODEs with constant coefficients

\[ay'' + by' + cy = f(t)\]

The characteristic equation $ar^2 + br + c = 0$ determines solution structure:

• Two real roots $r_1 \neq r_2$: $y_h = C_1 e^{r_1 t} + C_2 e^{r_2 t}$
• Repeated root: $y_h = (C_1 + C_2 t)e^{rt}$
• Complex roots $\alpha \pm i\beta$: $y_h = e^{\alpha t}(C_1\cos\beta t + C_2\sin\beta t)$

Stability: Lyapunov’s method

For the system $\dot{\mathbf{x}} = f(\mathbf{x})$, an equilibrium $\mathbf{x}^*$ is stable if there exists a Lyapunov function $V(\mathbf{x}) > 0$ with $\dot{V}(\mathbf{x}) \leq 0$.