Common differential equations
Exponential growth or decay
\[\frac{dN}{dt}=rN,\qquad N(t)=N_0e^{rt}.\]Logistic growth
\[\frac{dN}{dt}=rN\left(1-\frac NK\right).\]Linear loss with input
\[\frac{dx}{dt}=u(t)-kx.\]Diffusion equation
\[\frac{\partial u}{\partial t}=D\nabla^2u.\]Reaction–diffusion equation
\[\frac{\partial u}{\partial t}=f(u)+D\nabla^2u.\]General autonomous system
\[\frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x}).\]Equilibria satisfy \(\mathbf f(\mathbf x^*)=0\), and local stability is commonly studied using the Jacobian matrix.