Differential Equations Slope Field & Runge–Kutta (RK4) Phase Explorer

Q: Why is 4th-order Runge–Kutta (RK4) dramatically more accurate than Forward Euler?

Forward Euler extrapolates using only the left-endpoint slope with global error O(h), whereas RK4 blends four weighted trial slopes (k₁, 2k₂, 2k₃, k₄)/6 to achieve global error O(h⁴).

Click anywhere in the phase plane to launch multi-seed integral curves and compare 1st-order Euler drift against 4th-order Runge–Kutta accuracy.

Normalized Quiver Slope Field (Click Anywhere on Plane to Drop New RK4 Trajectory)

12 Canonical First-Order ODE & Phase-Stability Models

Thermal & First-Order Linear ODEs

Newton’s Law of Cooling dy/dt = −k(y − T_env) — Exponential Attractor Line

All trajectories exponentially approach horizontal equilibrium asymptote y = 1 regardless of initial temperature y(0).

dy/dx = -0.6*(y - 1)

Population Dynamics & Equilibria

Verhulst Logistic Differential Equation dy/dt = r·y(1 − y/K) — Phase Stability

Autonomous phase portrait contrasting unstable extinction equilibrium y = 0 against stable carrying capacity y = 3.

dy/dx = 0.8*y*(1 - y/3)

Nonlinear Bifurcation & Memory

Bistable Cubic ODE dy/dt = α·y(4 − y²) — Two Attractors & Separatrix Threshold

Initial states y(0) > 0 snap to upper attractor y = +2 while y(0) < 0 snap to y = −2, modeling Schmitt triggers and genetic switches.

dy/dx = 0.25*y*(4 - y^2)

Electrical Transients & Steady State

AC-Driven RC Low-Pass Filter ODE dy/dt + (1/RC)y = A·cos(ωt)

Shows transient initial offset decaying away until all RK4 trajectories lock onto the same phase-lagged sinusoidal steady state.

dy/dx = 2*cos(1.8*x) - 0.9*y

Separable Differential Equations

Separable ODE dy/dx = −kxy — Family of Gaussian Bell Solutions y = C·e^(−kx²/2)

Compares Euler’s 1st-order drift against 4th-order Runge–Kutta (RK4) across symmetric Gaussian peak trajectories.

dy/dx = -0.6*x*y

Nonlinear Riccati Equations

Riccati Nonlinear Differential Equation dy/dx = a·x² − b·y² (Airy Reduction)

Non-separable quadratic ODE appearing in optimal LQR control theory and WKB quantum tunneling approximations.

dy/dx = 0.25*x^2 - 0.35*y^2

Ecological Modelling & Tipping Points

Logistic Population with Seasonal Harvesting dy/dt = y(1 − y/K) − H(t)

Reveals seasonal limit-cycle oscillation above critical biomass vs. sudden population collapse below the saddle-node separatrix.

dy/dx = y*(1 - y/3) - 0.55*(1 + 0.6*sin(2*x))

Oncology & Actuarial Growth

Gompertz Asymmetric Growth ODE dy/dt = α·y·ln(K/y) — Inflection at K/e

Unlike symmetric logistic growth (inflection at K/2), Gompertz growth inflects earlier at y = K/e ≈ 0.368K.

dy/dx = 0.55*y*ln(3.5 / max(0.05, abs(y)))

Bernoulli Differential Equations

Bernoulli Cubic ODE dy/dx − P(x)y = Q(x)y³ — Exact Linearization Benchmark

Transforms into a linear ODE via substitution u = y^(1−n) = y⁻², providing an exact closed-form check on RK4 numerical precision.

dy/dx = 0.5*y - 0.2*y^3

Variable-Coefficient Linear ODEs

Integrating Factor ODE dy/dx + y/(x + 4) = cos(x)

Multiplying by μ(x) = x + 4 yields exact derivative d/dx[(x+4)y] = (x+4)cos(x), solved alongside interactive RK4 trajectories.

dy/dx = cos(x) - y / (x + 4)

Isoclines & Asymptotic Funnels

Parabolic Nullcline Funnel dy/dx = x − αy² — Fence & Funnel Theorem

Demonstrates Hubbard’s antifunnel/funnel theorem as trajectories converge tightly onto the upper parabolic branch y ≈ √(x/α).

dy/dx = x - 0.35*y^2

Nonlinear Mechanics

Sinusoidal Force & Linear Drag ODE dy/dx = −A·sin(x) − γ·y

Illustrates periodic potential valleys with viscous damping pulling trajectories into stable phase basins.

dy/dx = -1.5*sin(x) - 0.35*y

Mathematical Reference & Frequently Asked Questions

Why is 4th-order Runge–Kutta (RK4) dramatically more accurate than Forward Euler?

Forward Euler extrapolates using only the left-endpoint slope with global error O(h), whereas RK4 blends four weighted trial slopes (k₁, 2k₂, 2k₃, k₄)/6 to achieve global error O(h⁴).