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Differential Equations Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in differential equations.

Exact & Linear Differential Equations - Theory & Concepts

Solving First-Order DEs using Exactness, Integrating Factors, and Linear Methods (including Bernoulli's Equation).

Exact Equation Solution Visualizer

Potential Function:

$F(x,y) = x^2 y - x^3 - y^2 = C$

Adjust the constant $C$ to see different level curves (solutions) of the exact equation $(2xy - 3x^2)dx + (x^2 - 2y)dy = 0$.

C = 0.0

Applications of First-Order DEs - Theory & Concepts

Practical applications including population growth, decay, Newton's Law of Cooling, mixing problems, electrical circuits, and falling bodies.

Mixing Problem Simulator

Parameters

5 L/min
0.5 kg/L
5 L/min
Solute Amount
10.00 kg
Time
0.0 min
Inflow: 5 L/min
Outflow: 5 L/min
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Higher-Order Homogeneous DEs - Theory & Concepts

Solving nth-order linear homogeneous differential equations with constant coefficients and Cauchy-Euler equations.

Spring-Mass-Damper System

Parameters

1 kg
0.5 Ns/m
10 N/m
1 m
System State
Underdamped
Crit. Damping = 6.32
m
Eq
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Applications of Higher-Order DEs - Theory & Concepts

Practical applications in mechanical and electrical systems, including Spring-Mass Systems, Pendulums, and RLC Circuits.

Spring-Mass-Damper System

Parameters

1 kg
0.5 Ns/m
10 N/m
1 m
System State
Underdamped
Crit. Damping = 6.32
m
Eq
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Systems of Differential Equations - Theory & Concepts

Solving systems of linear differential equations using elimination, matrix methods (eigenvalues), and phase portrait analysis.

Phase Portrait Visualizer

System Matrix A

TypeNode
StabilityUnstable (Source)

Click on the plane to start a trajectory from that point.

Laplace Transforms - Theory & Concepts

Solving initial value problems using the Laplace Transform method, partial fraction decomposition, step functions, and Dirac delta.

Visualizing Convolution Integral

The convolution (f * g)(t) = \int_0^t f(\tau)g(t-\tau) d\tau involves flipping g, shifting it by t, multiplying it with f, and integrating the overlapping area.

Current Value of Convolution
(f * g)(2.0) \approx 0.822
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Stationary f(\tau)
Flipped/Shifted g(t-\tau)
Product (Area = Convolution)
Current t

Series Solutions - Theory & Concepts

Using power series methods, including Radius of Convergence and the Frobenius Method, to solve differential equations with variable coefficients.

Power Series Convergence Visualization

sin⁡(x)≈∑n=00(−1)nx2n+1(2n+1)!\sin(x) \approx \sum_{n=0}^{0} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
True Function
Approximation (1 term)
1 TermTerms: 115 Terms

Numerical Methods for DEs - Theory & Concepts

Solving differential equations using numerical approximations, including Euler's method, Runge-Kutta methods, and error analysis.

Direction Field & Euler's Method Simulator

Euler Formula:

yn+1=yn+h⋅f(xn,yn)y_{n+1} = y_n + h \cdot f(x_n, y_n)

Partial Differential Equations - Theory & Concepts

Introduction to PDEs, classification, Separation of Variables, and solving the Heat, Wave, and Laplace equations.

Fourier Series Visualization

f(t)≈4π∑k=11sin⁡((2k−1)t)2k−1f(t) \approx \frac{4}{\pi} \sum_{k=1}^{1} \frac{\sin((2k-1)t)}{2k-1}
1 Harmonicn = 120 Harmonics