dx1dt

dx1/dt denotes the time derivative of the first state variable x1 in a dynamical system. If the state vector is x(t) = [x1(t), x2(t), ..., xn(t)]^T, its evolution is written as dx/dt = f(x, t). The first component satisfies dx1/dt = f1(x, t). The notation dx1/dt or ẋ1 is standard in physics and engineering.

Interpretation: dx1/dt describes how x1 changes with time, depending on the current state x and possibly on time or inputs. If x1 represents a position, dx1/dt is velocity; if x1 is a concentration or temperature, it is the rate of change due to production, decay, or transport. The specific form of dx1/dt is determined by the model and may involve other state variables, external inputs, or explicit time dependence.

Examples: In a linear system dx/dt = A x + B u with x = [x1, x2]^T and u an input, the first equation is dx1/dt = a11 x1 + a12 x2 + b1 u. In a simple decay process where x1 is a substance amount, dx1/dt = -k x1.

Solving: To determine x1(t), an initial condition x1(t0) = x1_0 is required. Numerical methods such as Euler or Runge–Kutta approximate dx1/dt to advance x1 in time. dx1/dt is a central concept in state-space representations across science and engineering.