f(x) GraphLab 3D 2D/3D Graphing Studio, Step-by-Step Matrix & Symbolic Calculus Workbench العربية

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.

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

Initial-Value Trajectory Summary (RK4 vs. Euler)

dy/dxcos(x) - y / (x + 4) , y(-2) = 1.5
RK4 y(6)0.340076 (Euler: 0.23255)

More ODE Slope Field 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