∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 1 Theory

First-Order ODEs, Slope Fields & Autonomous Bifurcations

RK4 trajectory integration, separable/linear ODEs, and equilibria

Drag the initial condition (x₀, y₀) across the 2D slope field dy/dx = F(x,y) and watch the Runge-Kutta (RK4) solver trace the exact integral curve in real time.

1. Slope Fields & Autonomous Equilibria

A first-order ODE dy/dx = F(x,y) assigns a tangent slope to every point in the plane. For an autonomous equation dy/dt = f(y), roots where f(y*) = 0 form horizontal equilibrium lines—attractors if f'(y*) < 0 and repellers if f'(y*) > 0.

dydt=y(2−y)  ⟹  y∗=0 (unstable),  y∗=2 (stable attractor)\frac{dy}{dt} = y(2 - y) \implies y^* = 0 \text{ (unstable)},\; y^* = 2 \text{ (stable attractor)}

Quick Reference & Cheat Sheet

Integrating Factor for Linear 1st-Order ODEs
y′+P(x)y=Q(x)  ⟹  μ(x)=e∫P(x) dx,y=1μ∫μQ dxy' + P(x)y = Q(x) \implies \mu(x) = e^{\int P(x)\,dx},\quad y = \frac{1}{\mu}\int \mu Q\,dx
Multiplies the ODE so the left-hand side collapses via the Product Rule.