First derivative: rate of change (velocity, growth rate)

Second derivative: change of rate (acceleration, curvature)

Initial Value Problems (IVPs): initial state predicts future behavior

Boundary Value Problems (BVPs): constraints at multiple points fit behavior over an interval

Examples: y(t), x(t), P(t), T(x,t)

Time t, space x, or multiple variables (x,y,z,t)

Ordinary derivatives (ODE): dy/dt, d²y/dt²

Partial derivatives (PDE): ∂u/∂t, ∂²u/∂x²

Constants: mass m, spring constant k, growth rate r

Forcing terms: external input / driving signal

One independent variable

Example form: dy/dt = f(t, y)

Two or more independent variables

Example form: ∂u/∂t = F(x, t, u, ∂u/∂x, …)

Linear

Nonlinear

Highest derivative present (first-order y', second-order y'', higher-order y^(n))

Autonomous: dy/dt = f(y) (no explicit t)

Non-autonomous: dy/dt = f(t, y)

Deterministic: no randomness

Stochastic: includes noise terms (common in finance, biology)

General solution: family of solutions with constants

Particular solution: satisfies given conditions

Closed-form (exact expressions)

Numerical approximations (Euler, Runge–Kutta, finite difference/element)

Under suitable conditions, an IVP has a unique solution

Specify value at a starting point (e.g., y(t0) = y0)

For an nth-order ODE: need n conditions (y, y', …, y^(n−1) at t0)

Conditions at different points (e.g., y(0)=0 and y(L)=1)

Conditions encode starting state or geometry/physics constraints

Choose state variable(s); decide independent variable(s) (time/space)

Decide what effects are included/ignored (homogeneity, constants, small-angle, etc.)

Accumulation = Inflow − Outflow + Generation − Consumption

Hooke, Newton cooling, Fick diffusion, Ohm law

Fit to data; compare predictions with observations

Equilibria, stability, time scales, sensitivity

Model: dP/dt = rP

Meaning: change proportional to current amount

Solution: P(t)=P0 e^{rt}

Applications: population (idealized), radioactive decay, interest

Model: dP/dt = rP(1 − P/K)

Meaning: growth slows as P approaches carrying capacity K

Features: equilibria at P=0 and P=K; S-shaped curve

Model: dT/dt = −k (T − T_env)

Meaning: change proportional to difference from environment

Model: m x''(t) = F(x, x', t)

Examples: free fall with drag m x'' = mg − c x'; spring-mass m x'' + kx = 0; damped m x'' + c x' + kx = 0

Model: C dV/dt + (1/R) V = (1/R) V_in(t)

Meaning: capacitor voltage follows charging/discharging dynamics

Model: ∂u/∂t = D ∂²u/∂x²

Meaning: spreading driven by curvature/gradients

Model: ∂²u/∂t² = c² ∂²u/∂x²

Meaning: disturbances propagate at speed c

Visualize solution flow; equilibria where dy/dt = 0

Stable: nearby solutions return

Unstable: nearby solutions diverge

Fast vs slow dynamics; stiffness in numerics

Energy-like quantities constant in time for some systems

Euler / Improved Euler

Runge–Kutta (RK4)

Finite difference / finite element for PDEs