ODE Solver

Direction fields, Euler/RK4 methods, and phase planes.

Parameters

dy/dx = f(x,y)

x min

x max

y min

y max

Initial Condition

x₀

y(x₀) = y₀

dy/dx = f(x,y)

x₀

y₀

Step h

Target x

Euler y —

RK4 y —

dx/dt = f(x,y)

dy/dt = g(x,y)

f(x) (the x part)

g(y) (the y part)

x	Euler y	RK4 y

Ordinary Differential Equations (ODE) Handbook

Mathematical Principles & Theorems

Simulates Initial Value Problems \(\frac{d\mathbf{y}}{dt} = \mathbf{f}(t, \mathbf{y}), \mathbf{y}(t_0) = \mathbf{y}_0\). Numerical time-stepping integrators: (1) Forward Euler: \(\mathbf{y}_{n+1} = \mathbf{y}_n + h\mathbf{f}(t_n, \mathbf{y}_n)\) (first-order \(\mathcal{O}(h)\)); (2) Runge-Kutta 4th Order (RK4): \(\mathbf{y}_{n+1} = \mathbf{y}_n + \frac{h}{6}(\mathbf{k}_1 + 2\mathbf{k}_2 + 2\mathbf{k}_3 + \mathbf{k}_4)\) (fourth-order \(\mathcal{O}(h^4)\)). 2nd-order nonlinear systems (Damped Pendulum, Lotka-Volterra Predator-Prey, Van der Pol oscillator) analyzed in 2D phase space \((x, \dot{x})\).

Operating Instructions

Select an ODE system preset (Harmonic Oscillator, Lotka-Volterra, Damped Pendulum, Logistic) or custom equations.
Set step size \(h\), integration duration, and initial conditions \((x_0, y_0)\) using numeric inputs.
Click Simulate to compute numerical trajectories via RK4 integration.
Inspect the phase portrait canvas showing vector field streamlines, nullclines, and limit cycles.