Direction Field
\(\dfrac{dy}{dx} = f(x,y)\)
Visualize slope fields and overlay solution curves for first-order ODEs. Enter any function \(f(x,y)\) and see the direction field update in real time.
ExploreDifferential Equations
First-order ODEs, oscillations, integral transforms, and phase-plane analysis. We'd love to hear what you think. .
This collection covers fundamental topics in ordinary differential equations (ODEs): first-order equations, second-order linear systems, and integral transforms for solving ODEs.
First-Order Equations
- Direction Field — plot slope fields for \(\dfrac{dy}{dx} = f(x,y)\) and overlay solution curves.
- Phase Line Plot — analyze equilibrium points and stability for autonomous ODEs \(\dfrac{dy}{dt} = f(y)\).
- Newton's Law of Cooling — solve \(\dfrac{dT}{dt} = -k(T - T_a)\) with a visual heat-transfer simulation.
Second-Order & Higher-Order Equations
- Mass-Spring-Damping — simulate \(m\ddot{x} + c\dot{x} + kx = 0\) across underdamped, critically damped, and overdamped regimes.
- Mass-Spring (Multiple DOF) — coupled mass-spring systems and normal modes of vibration.
- Forced Vibrations — transient and steady-state behavior including resonance via \(m\ddot{x} + c\dot{x} + kx = F(t)\).
- Laplace Transforms — visualize \(\mathcal{L}\{f(t)\}(s)=\int_0^\infty f(t)e^{-st}\,dt\) as a signed area integral with shifting theorems.
- Convolution — watch \(g(t-\tau)\) slide over \(f(\tau)\) and see \(h(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau\) build in real time.
Systems & Phase Analysis
- Linear Phase Portraits — classify fixed points of \(\mathbf{x}' = A\mathbf{x}\) via eigenvalue analysis.
- Nonlinear Phase Portraits — explore nonlinear vector fields and stability via linearization.
Accessibility
All tools meet WCAG 2.1 AA standards: keyboard navigation, screen reader support via MathJax assistive MathML, high-contrast mode, visible focus indicators, and ARIA labels throughout.
Tool Descriptions
Phase Line Plot
\(\dfrac{dy}{dt} = f(y)\)
Analyze equilibrium points and stability for autonomous first-order ODEs. Identify stable, unstable, and semi-stable fixed points on the phase line.
ExploreMass-Spring-Damping Simulator
\(m\ddot{x} + c\dot{x} + kx = 0\)
Simulate underdamped, critically damped, and overdamped motion. Observe displacement and phase-plane trajectories in real time.
ExploreMultiple Degree of Freedom Mass Spring
\(M\ddot{\mathbf{x}} + C\dot{\mathbf{x}} + K\mathbf{x} = \mathbf{0}\)
Simulate coupled mass-spring systems and explore normal modes of vibration. Visualize how energy transfers between coupled oscillators.
ExploreForced Vibrations and Resonance
\(m\ddot{x} + c\dot{x} + kx = F_0\cos(\omega t)\)
Observe transient and steady-state behavior, including resonance. Explore how driving frequency affects amplitude response.
ExploreLaplace Transform Visualizer
\(\mathcal{L}\{f(t)\}(s) = \displaystyle\int_0^{\infty} f(t)\,e^{-st}\,dt = F(s)\)
Visualize the Laplace transform as a signed area integral. Watch the weighted integrand accumulate, track convergence, and explore the first and second shifting theorems.
ExploreConvolution Visualizer
\((f*g)(t) = \displaystyle\int_0^{t} f(\tau)\,g(t-\tau)\,d\tau\)
Watch \(g(t-\tau)\) slide over \(f(\tau)\) as \(t\) grows. The overlapping area accumulates into \(h(t)\), tracing out the exact convolution in real time.
ExploreLinear Phase Portraits
\(\mathbf{x}' = A\mathbf{x}, \quad A \in \mathbb{R}^{2\times 2}\)
Visualize trajectories for 2D linear systems based on eigenvalue classification — nodes, spirals, saddles, and centers.
ExploreNonlinear Phase Portraits
\(x' = f(x,y), \quad y' = g(x,y)\)
Explore vector fields and nullclines for nonlinear autonomous systems. Identify fixed points and their local stability via linearization.
ExploreNewton's Law of Cooling
\(\dfrac{dT}{dt} = -k(T - T_a)\)
Simulate heat transfer with realistic object visuals, jet colormap temperature display, steam effects, and real-time numerical solution.
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