Interactive Dynamics
Welcome to this exploration of dynamics, the mathematics of systems that change over time. This module uses interactive tools to make the core concepts of differential equations intuitive and tangible. You will see how discrete changes evolve into continuous models and learn to describe, analyze, and apply these powerful mathematical ideas.
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Learning Objectives
- Translate verbal descriptions of dynamic phenomena into the language of differential equations.
- Understand the relationship between discrete recurrence relations and continuous differential equations.
- Analyze the qualitative behavior of solutions using slope fields and equilibrium points.
- Approximate functions and solve differential equations using Taylor series expansions.
- Apply differential equation models to real-world scenarios in finance, medicine, and population dynamics.
Continuous Change
Many systems can be modeled with discrete steps, like monthly compound interest. As the time between steps shrinks (i.e., the number of compounding periods $n$ increases), these discrete models approach a continuous model. The chart below shows the final amount after one year, $A = P(1 + r/n)^n$, as $n$ increases. Notice how it converges to the continuous limit of $P \cdot e^{rt}$.
Amount vs. Periods (n)
Growth Over Time
| Period | Start Value | End Value |
|---|
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Qualitative Analysis: Slope Fields
Even without solving a differential equation, we can understand its solutions' behavior. The "Phase Plot" below shows the rate of change (the derivative) for any given value of the variable. The "Slope Field" shows the corresponding slopes. Hover over the slope field to see the derivative value and its projection on the phase plot. Click on the slope field to draw a solution curve.
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Taylor Series Approximations
Many complex functions can be approximated by simpler polynomials called Taylor polynomials. As we add more terms (increasing the order of the polynomial), the approximation becomes more accurate near the center point. Use the slider to change the order of the Taylor polynomial and see how it improves.
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Solving ODEs
Differential equations are the language of change. They express a relationship between a quantity and its rate of change. Select a real-world scenario to see its corresponding differential equation and how to solve it.
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Modeling Population Growth
The logistic equation models population growth that is limited by environmental factors. The population grows exponentially at first but levels off as it approaches the carrying capacity ($N$). Adjust the growth rate ($k$) and carrying capacity to see how they affect the population curve over time.
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Modeling Mortgage
A mortgage balance can be modeled with a differential equation. The balance ($A$) increases due to interest and decreases with payments. The rate of change is $dA/dt = rA - W$, where $r$ is the annual interest rate and $W$ is the total annual payment. For a fixed initial loan of $300,000, adjust the sliders to see how these factors affect the time it takes to pay off the loan.
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Modeling Infectious Disease: The SI Model
The spread of a disease in a population where individuals do not recover can be modeled by the Susceptible-Infected (SI) model. The population is divided into two groups: those who are susceptible ($S$) and those who are infected ($I$). The total population is $N = S + I$.
The rate at which people get infected depends on the number of infected people and the number of susceptible people. This gives us the differential equation:
$$ \frac{dI}{dt} = \beta I S = \beta I (N - I) $$
Here, $\beta$ (beta) is the transmission rate, which combines the contact rate and the probability of transmission. This is a form of the logistic equation.
Case Study: "28 Days Later"
In the movie "28 Days Later," a virus spreads through London. Imagine a protagonist wakes from a coma after 28 days to find the city seemingly empty, implying almost everyone has been infected. We can use our model to estimate the transmission rate ($\beta$) that would lead to such a rapid outbreak.
Let's assume the population of London is $N = 9,000,000$ and the outbreak starts with one person, $I(0) = 1$. Your challenge is to find the value of $\beta$ that results in 99% of the population being infected in approximately 28 days.
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Modeling Pendulum Dynamics
The equation of motion is derived from Newton's second law, $F=ma$. The restoring force perpendicular to the rod is $F = -mg \sin(\theta)$, and the tangential acceleration is $a = L \frac{d^2\theta}{dt^2}$. This leads to the following second-order differential equation:
$$ \frac{d^2\theta}{dt^2} + \frac{g}{L} \sin(\theta) = 0 $$
To analyze this, we convert it into a system of two first-order equations by defining angular velocity $\omega = d\theta/dt$:
- $d\theta/dt = \omega$
- $d\omega/dt = -(g/L) \sin(\theta)$
The "Phase Portrait" shows the vector field of the system, indicating the direction of motion at any point in the phase space. The "Phase Space Plot" shows the specific trajectory (path) of the pendulum. Click on either plot to start the simulation from that state.
Pendulum Animation
Phase Space Plot
Phase Portrait
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