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Differential Equations Simulations

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

First-Order DEs (Separable & Homogeneous) - Theory & Concepts - First Order Separable And Homogeneous

Introduction to Differential Equations, Initial Value Problems, Variable Separable method, and Homogeneous Differential Equations.

Equation Type

Initial Condition (x0,y0)(x_0, y_0)

1.0
1.0

Drag the sliders to change the initial point (x0,y0)(x_0, y_0) and see how the particular solution follows the direction field.

Exact & Linear Differential Equations - Theory & Concepts - Exact And Linear Equations

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

Linear DE

y′+2xy=4xy' + \frac{2}{x}y = 4x

Find an integrating factor μ(x)=xk\mu(x) = x^k to make it exact.

Select power kk

0.0

✗ Not Exact

μ(x)=x0\mu(x) = x^{0}
ddxμ(x)\frac{d}{dx}\mu(x)
P(x)μ(x)P(x)\mu(x)

To be exact, we need dμdx=P(x)μ(x)\frac{d\mu}{dx} = P(x)\mu(x). Match the blue curve to the dashed red curve.

Applications of First-Order DEs - Theory & Concepts - Applications Of First Order Des

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

Mixing Tank Model

dAdt=RinCin−RoutAV(t)\frac{dA}{dt} = R_{in} C_{in} - R_{out} \frac{A}{V(t)}
3 L/min
2 kg/L
3 L/min
Salt Amount10.0 kg
Tank Volume100.0 L
A(t) vs t

Higher-Order Homogeneous DEs - Theory & Concepts - Higher Order Homogeneous Equations

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

Spring-Mass System

my′′+cy′+ky=0m y'' + c y' + k y = 0
Underdamped (Complex Conjugate Roots)
1.0
0.5
4.0
Auxiliary Roots
r=−0.25±1.98ir = -0.25 \pm 1.98i
m
y(t) vs t

Higher-Order Non-Homogeneous DEs - Theory & Concepts - Higher Order Non Homogeneous Equations

Solving non-homogeneous linear differential equations using Undetermined Coefficients and Variation of Parameters.

Forced Vibrations

y′′+4y=cos⁡(ωt)y'' + 4y = \cos(\omega t)

Explore the superposition principle: y(t)=yh(t)+yp(t)y(t) = y_h(t) + y_p(t). Observe resonance when forcing frequency ω\omega matches natural frequency ω0=2\omega_0 = 2.

1.00 rad/s

Toggle Components

max_y ≈ 1.0

Applications of Higher-Order DEs - Theory & Concepts - Applications Of Higher Order Des

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

RLC Circuit Model

Ld2qdt2+Rdqdt+1Cq=0L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = 0
1.0 H
1.0 Ω\Omega
2.0 F⁻¹
Charge q(t)
Time t = 0.0s

Systems of Differential Equations - Theory & Concepts - Systems Of Des

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

Phase Portrait

(x′y′)=(−1.0−1.01.0−1.0)(xy)\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -1.0 & -1.0 \\ 1.0 & -1.0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}
Critical Point (0,0): Stable Spiral
-1.0
-1.0
1.0
-1.0
Tr(A)=−2.00\text{Tr}(A) = -2.00
det⁡(A)=2.00\det(A) = 2.00
Δ=Tr2−4det⁡=−4.00\Delta = \text{Tr}^2 - 4\det = -4.00
x range: [-5, 5]

Laplace Transforms - Theory & Concepts

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

Laplace Transform Pairs

Select a function to see its representation in the time domain tt and the complex frequency domain ss.

Transform Pair
L{e−at(a=1)}=1s+1\mathcal{L}\{e^{-at} \quad (a=1)\} = \frac{1}{s+1}

Time Domain f(t)f(t)

t = [0, 10]

s-Domain F(s)F(s)

s = [0, 5]

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 Solution

Solving y′=yy' = y with y(0)=1y(0)=1 around the ordinary point x0=0x_0=0.

y(x)≈∑n=00xnn!y(x) \approx \sum_{n=0}^{0} \frac{x^n}{n!}

Notice how adding more terms increases the interval where the polynomial approximation closely matches the exact exponential solution.

1
LinearDegree 9
Exact Solution exe^x
Series Approximation
x ∈ [-3, 3]

Numerical Methods for DEs - Theory & Concepts

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

Numerical Methods Comparison

Solving y′=yy' = y with y(0)=1y(0)=1.

Compare the simple Euler method (1st order) to the Runge-Kutta method (4th order). Notice how much faster RK4 converges to the exact solution as step size hh decreases.

1.00
Fine (0.1)Coarse (1.5)
Exact (exe^x)
Euler's MethodE ~ O(h)
RK4 MethodE ~ O(h^4)
x ∈ [0, 3]

Partial Differential Equations - Theory & Concepts

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

1D Wave Equation

∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Vibrating string fixed at both ends (x=0x=0, x=Lx=L). The solution is a superposition of normal modes (standing waves).

1

Determines the spatial frequency sin⁡(nπx/L)\sin(n\pi x/L).

1.0
u(x,t)
un(x,t)=sin⁡(1πxL)cos⁡(1π(1.0)tL)u_n(x,t) = \sin\left(\frac{1\pi x}{L}\right) \cos\left(\frac{1\pi (1.0) t}{L}\right)