MTH 114M - Ordinary Differential Equations Learning Platform

ODEs in Practice

Real-World Applications of Differential Equations

Differential equations model change in physical, biological, economic, and engineering systems. Explore interactive demonstrations in the section below.

Electrical Circuits

Circuits with resistors, capacitors, and inductors follow first and second-order ODEs describing how voltage and current change over time.

RC Circuit: \(RC\frac{dV}{dt} + V = V_0\)

RL Circuit: \(L\frac{dI}{dt} + RI = V(t)\)

RLC Circuit: \(L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{q}{C} = E(t)\)

Mechanics

Newton's laws lead to ODEs for motion. Drag forces, springs, and pendulums all exhibit characteristic ODE behavior.

Free Fall with Drag: \(m\frac{dv}{dt} = mg - kv^2\)

Damped Spring: \(m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0\)

Small-angle Pendulum: \(\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0\)

Heat Transfer

Temperature changes follow ODEs when heat flows between objects. Newton's law of cooling is a classic first-order linear ODE.

Newton's Cooling: \(\frac{dT}{dt} = -k(T - T_{env})\)

Radiative Cooling: \(\frac{dT}{dt} = -\sigma(T^4 - T_0^4)\)

Heating with Source: \(\frac{dT}{dt} = -k(T - T_{env}) + Q(t)\)

Population Dynamics

Population growth models range from simple exponential to logistic equations with carrying capacity.

Exponential Growth: \(\frac{dP}{dt} = kP\)

Logistic Growth: \(\frac{dP}{dt} = rP(1 - \frac{P}{K})\)

Harvesting: \(\frac{dP}{dt} = rP(1 - \frac{P}{K}) - H\)

Medicine & Pharmacology

Drug concentration in the bloodstream follows first-order kinetics. Epidemiological models use systems of ODEs to track disease spread.

Drug Elimination: \(\frac{dC}{dt} = -kC\)

Drug with Dosing: \(\frac{dC}{dt} = -kC + D(t)\)

Tumor Growth: \(\frac{dV}{dt} = aV\ln(\frac{K}{V})\)

Nuclear & Chemistry

Radioactive decay and chemical reactions follow rate laws that are naturally expressed as first-order or higher-order ODEs.

Radioactive Decay: \(\frac{dN}{dt} = -\lambda N\)

Decay Chain: \(\frac{dN_1}{dt} = -\lambda_1 N_1\), \(\frac{dN_2}{dt} = \lambda_1 N_1 - \lambda_2 N_2\)

Second-order Reaction: \(\frac{d[A]}{dt} = -k[A]^2\)

Economics & Finance

Compound interest, loan amortization, and price dynamics are modeled with first-order linear and separable equations.

Compound Interest: \(\frac{dA}{dt} = rA\)

Loan with Payments: \(\frac{dP}{dt} = rP - m\)

Price Adjustment: \(\frac{dp}{dt} = k(D(p) - S(p))\)

Fluid Mixing

Concentration changes in tanks with inflow and outflow follow first-order linear ODEs, classic examples in engineering courses.

Single Tank: \(\frac{dS}{dt} = r_{in}c_{in} - \frac{r_{out}S}{V}\)

Variable Volume: \(\frac{d(VS)}{dt} = r_{in}c_{in} - r_{out}\frac{S}{V}\)

Atmospheric Pressure: \(\frac{dP}{dh} = -\frac{Mg}{RT}P\)

Interactive Demonstrations

Click on any demonstration to explore differential equations in action.

First-Order ODE Applications

Population Growth

Interactive

Logistic equation: \(\frac{dP}{dt} = rP(1 - P/K)\)

Explore exponential and logistic growth models with sliders.

Radioactive Decay

Interactive

Exponential decay: \(\frac{dN}{dt} = -\lambda N\)

Visualize decay curves and half-life calculations.

RC Circuits

Interactive

Circuit equation: \(RC\frac{dV}{dt} + V = V_0\)

First-order ODE: Analyze voltage changes in circuits.

Second-Order & Variable Coefficient ODEs

RLC Circuits

Interactive

Circuit equation: \(L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{q}{C} = V_0\)

Second-order ODE: Explore damped oscillations.

Bessel Functions

Interactive

Bessel's equation: \(x^2y'' + xy' + (x^2 - n^2)y = 0\)

Variable coefficients: Vibrating membranes & heat conduction.

Airy Functions

Interactive

Airy's equation: \(y'' - xy = 0\)

Variable coefficients: Quantum mechanics & optics.

ODEs from Partial Differential Equations

Heat Equation

Interactive

Separation yields: \(T' = -\lambda kT\) and \(X'' + \lambda X = 0\)

See how separation of variables reduces PDE to ODEs.

Wave Equation

Interactive

Separation yields: \(T'' + c^2\lambda T = 0\) and \(X'' + \lambda X = 0\)

Vibrating string: Two coupled ODEs from one PDE.

Why ODE Models? Physics-based vs. Data-driven Approaches

Key Insight: While data-driven models (polynomial fits, neural networks) can match training data, ODE models capture underlying physics and generalize better.

Extrapolation: ODE models predict beyond training data range; curve fits often fail outside observed data
Physical meaning: ODE parameters (drag coefficient, decay constant) have interpretable units and physical significance
Data efficiency: ODE models work with sparse data; neural networks need thousands of samples
Conservation laws: ODE models naturally respect energy/mass conservation; data models may violate physics

Planetary Orbits

Interactive

Compare: Polynomial fit vs. \(\frac{d^2r}{dt^2} = -\frac{GM}{r^2}\)

A polynomial fit matches known positions but fails to predict beyond observation window. The ODE model predicts entire orbit from any initial condition.

Population Growth

Interactive

Compare: Polynomial fit vs. logistic ODE \(\frac{dP}{dt} = rP(1-P/K)\)

Polynomial fits early growth data but predicts unbounded growth. Logistic ODE correctly predicts carrying capacity saturation.

An Introductory Discussion

Understanding the fundamental concepts of ordinary differential equations.

a The k^th Order ODE

An ordinary differential equation (ODE) involves an unknown function y(x) and its derivatives with respect to a single independent variable x.

F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots, \frac{d^k y}{dx^k}\right) = 0

The order of an ODE is the highest derivative that appears. A k^th order ODE contains derivatives up to y^(k). For example, y'' + 3y' + 2y = 0 is second-order (k=2).

b Solutions of ODEs: Explicit, Implicit, and Formal

Explicit Solution

y expressed directly as a function of x:

y = Ce^{2x}

Implicit Solution

Relationship between y and x (real solutions exist):

\[x^2 + y^2 - C^2 = 0\]

Defines circles of radius |C|

Formal Solution

Relation yielding complex-valued explicit solutions:

\[x^2 + y^2 + C^2 = 0\]

No real solutions for C ≠ 0

c Do Solutions Always Exist?

Not all ODEs have solutions, and when they do, solutions may not be unique!

Existence Theorem (Peano)

If f(x,y) is continuous near (x₀, y₀), then y' = f(x,y) has at least one solution passing through that point.

Uniqueness (Picard-Lindelöf)

If ∂f/∂y is also continuous (Lipschitz condition), the solution is unique.

[None] No Solution Example:

\[|y'| + |y| + 1 = 0\]

Since |y'| ≥ 0, |y| ≥ 0, and 1 > 0, the sum is always ≥ 1, never zero.

[Many] Infinitely Many Solutions:

y' = 3y^{2/3}, \quad y(0) = 0

Has y = 0 and y = (x−c)³ for any c ≥ 0.

d Family of Solutions

The general solution of a k^th order ODE typically contains k arbitrary constants, representing a family of solution curves.

First-order ODE:

y' = y \implies y = Ce^x

One-parameter family (C)

Second-order ODE:

y'' + y = 0 \implies y = C_1\cos x + C_2\sin x

Two-parameter family (C₁, C₂)

A particular solution is obtained by specifying values for all constants, often determined by initial or boundary conditions.

e Initial Value Problems vs Boundary Value Problems

Initial Value Problem (IVP)

All conditions specified at a single point x = x₀:

y'' + y = 0, \quad y(0) = 1, \quad y'(0) = 0

Example: Position and velocity of a projectile at launch time.

Boundary Value Problem (BVP)

Conditions specified at different points:

y'' + y = 0, \quad y(0) = 0, \quad y(\pi) = 0

Example: Temperature at both ends of a rod.

Key Difference: IVPs usually have unique solutions (under mild conditions). BVPs may have no solution, one solution, or infinitely many solutions depending on the boundary conditions.

First Order Differential Equations

Higher Order Linear Differential Equations

Solve second and higher-order linear ODEs with various techniques.

Step 1: Select Equation Type

Form: \(a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0\)

Enter coefficients (highest order first, space-separated):

Examples:

Euler-Cauchy Equation: \(ax^2 y'' + bx y' + cy = 0\)

a (x²y''):

b (xy'):

c (y):

Examples:

Select Solution Method

Characteristic Equation

Recommended

For constant coefficient homogeneous equations. Find roots of characteristic polynomial.

\[ar^2 + br + c = 0\]

Euler-Cauchy Method

Coming Soon

For equations of form \(ax^2y'' + bxy' + cy = 0\). Use substitution \(y = x^r\).

Reduction of Order

Coming Soon

Given one solution \(y_1\), find second solution using \(y_2 = v(x) y_1\).

Variation of Parameters

Coming Soon

For non-homogeneous equations. Find particular solution using Wronskian.

Step-by-Step Solution

1 Write the Characteristic Equation

Replace \(y^{(n)}\) with \(r^n\) to get the characteristic polynomial:

Your characteristic equation:

Reference: Solution Forms by Root Type

Real Distinct Roots

Roots: \(r_1 \neq r_2\)

y = C_1 e^{r_1 x} + C_2 e^{r_2 x}

Complex Conjugate Roots

Roots: \(\alpha \pm \beta i\)

y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)

Repeated Roots

Root: \(r\) with multiplicity \(k\)

y = (C_1 + C_2 x + \cdots + C_k x^{k-1})e^{rx}

Select Application

Orthogonal Trajectories

Active

Find curves that intersect a given family at right angles (90°).

Examples: Isotherms & heat flow, equipotentials & field lines

Oblique Trajectories

Active

Find curves that intersect a given family at a fixed angle θ.

ODEs in Practice

Real-World Applications of Differential Equations

Electrical Circuits

Mechanics

Heat Transfer

Population Dynamics

Medicine & Pharmacology

Nuclear & Chemistry

Economics & Finance

Fluid Mixing

Interactive Demonstrations

First-Order ODE Applications

Population Growth

Radioactive Decay

RC Circuits

Second-Order & Variable Coefficient ODEs

RLC Circuits

Bessel Functions

Airy Functions

ODEs from Partial Differential Equations

Heat Equation

Wave Equation

Why ODE Models? Physics-based vs. Data-driven Approaches

Planetary Orbits

Population Growth

Interactive Demo

RC Circuit Analysis (First-Order ODE)

Circuit Diagram

The RC Circuit Equation

Parameters

Voltage vs Time

RLC Circuit Analysis (Second-Order ODE)

Circuit Diagram

The RLC Circuit Equation

Parameters

Charge & Current vs Time

An Introductory Discussion Lecture Slides

a The kth Order ODE

b Solutions of ODEs: Explicit, Implicit, and Formal

c Do Solutions Always Exist?

d Family of Solutions

e Initial Value Problems vs Boundary Value Problems

Initial Value Problem (IVP)

Boundary Value Problem (BVP)

First Order Differential Equations

Higher Order Linear Differential Equations

Step 1: Select Equation Type

Equation Confirmed

Select Solution Method

Characteristic Equation

Euler-Cauchy Method

Reduction of Order

Variation of Parameters

Step-by-Step Solution

Reference: Solution Forms by Root Type

Real Distinct Roots

Complex Conjugate Roots

Repeated Roots

Select Application

Orthogonal Trajectories

Oblique Trajectories

Instructor Interface

Module Availability

Predefined Examples

Slide Decks

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An Introductory Discussion

a The k^th Order ODE