AOS4 Topic 9: First Order Differential Equations

A differential equation contains derivatives of a particular function or variable. The following are examples of differential equations:

  1. \( \frac{dy}{dx} = \cos x \)
  2. \( \frac{d^2y}{dx^2} - 4 \frac{dy}{dx} = 0 \)
  3. \( \frac{dy}{dx} = \frac{y}{y + 1} \)

The solution of a differential equation is a clear definition of the function or relation, without any derivatives involved. For example, if \( \frac{dy}{dx} = \cos x \), then \( y = \int \cos x \, dx \) and so \( y = \sin x + c \), where \( c \) is a constant.


Professor Leonard

Watch the following video from Professor Leonard to learn ODES. This topic will be added more thoroughly soon. 


Basic Concepts in Differential Equations

  • Order of Differential Equations: The order of a differential equation is the highest derivative present in the equation.

  • Linear vs. Nonlinear Differential Equations: Linear differential equations have linear combinations of the dependent variable and its derivatives, while nonlinear equations do not.

  • Initial Value Problems (IVPs) and Boundary Value Problems (BVPs): IVPs involve finding a solution that satisfies an equation and initial conditions, while BVPs involve additional boundary conditions.


The logistic differential equation

The logistic differential equation is a fundamental model used to describe population growth that takes into account limiting factors such as carrying capacity. It is expressed as:

\[ \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \]

where:

  • \( \frac{dN}{dt} \) represents the rate of change of the population over time \( t \),
  • \( N \) is the population size at time \( t \),
  • \( r \) is the intrinsic growth rate of the population, and
  • \( K \) is the carrying capacity, which is the maximum population size that the environment can sustain indefinitely.

This equation captures the idea that the rate of population growth is proportional to the current population size \( N \), but it also decreases as \( N \) approaches the carrying capacity \( K \). When \( N \) is small compared to \( K \), the population grows exponentially. However, as \( N \) approaches \( K \), the growth rate slows down until the population stabilizes at \( K \).

The logistic differential equation is commonly used in ecology, biology, and other fields to model population dynamics in environments with limited resources or space. It provides insights into how populations interact with their environment and reach equilibrium over time.


Examples of Differential Equations

1. Biology

Population Growth

The logistic growth model for population dynamics is given by the differential equation:

\[ \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \]

where \(N\) is the population size, \(t\) is time, \(r\) is the intrinsic growth rate, and \(K\) is the carrying capacity.

Enzyme Kinetics

Enzyme-substrate interactions in biochemistry are often described by the Michaelis-Menten equation:

\[ \frac{d[S]}{dt} = \frac{V_{max}[S]}{K_m + [S]} - k_2[E][S] \]

where \([S]\) is the substrate concentration, \(t\) is time, \(V_{max}\) is the maximum reaction rate, \(K_m\) is the Michaelis constant, \(k_2\) is the rate constant for product formation, and \([E]\) is the enzyme concentration.


2. Chemistry

Chemical Reactions

Chemical kinetics involves differential rate equations to describe how reactants transform into products over time. For a simple reaction \(A + B \rightarrow C\), the rate of change of \(A\) is given by:

\[ \frac{d[A]}{dt} = -k[A][B] \]

where \([A]\) and \([B]\) are the concentrations of reactants, and \(k\) is the rate constant.


3. Economics

Supply and Demand

Economic models often use differential equations to represent supply and demand dynamics. For example, the supply-demand equilibrium can be described by:

\[ \frac{dS}{dt} = \alpha S - \beta D, \quad \frac{dD}{dt} = \gamma D - \delta S \]

where \(S\) is the supply, \(D\) is the demand, and \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) are parameters representing production, consumption, and market behavior.


4. Physics

Newton's Second Law

The motion of objects is governed by Newton's second law, which can be expressed as:

\[ F = m\frac{d^2x}{dt^2} \]

where \(F\) is the force acting on the object, \(m\) is its mass, and \(x\) is its position.



Analytical and Numerical Solutions of Differential Equations

Analytical Solution:
An analytical solution of a differential equation involves finding an explicit mathematical expression that represents the solution for all possible values of the independent variable. This solution is derived using mathematical techniques such as integration, separation of variables, and series expansions. Analytical solutions provide precise and exact solutions to differential equations.

For example, the analytical solution of a simple first-order ordinary differential equation like \( \frac{dy}{dt} = f(t, y) \) can be obtained by integrating both sides with respect to \( t \) and solving for \( y \). If the equation is linear, it may involve finding an integrating factor to simplify the integration process.

Numerical Solution:
A numerical solution of a differential equation involves approximating the solution using numerical methods, especially when analytical solutions are not feasible or too complex. Numerical methods discretize the domain of the problem and approximate the derivatives using finite differences or other techniques. Common numerical methods for solving differential equations include Euler's method, Runge-Kutta methods, finite difference methods, and finite element methods.

For example, to numerically solve the differential equation \( \frac{dy}{dt} = f(t, y) \) with initial condition \( y(t_0) = y_0 \), Euler's method iteratively updates the solution using the formula:
\[ y_{n+1} = y_n + hf(t_n, y_n) \]
where \( h \) is the step size and \( t_n = t_0 + nh \). This method approximates the solution by taking small steps in the domain.

In summary, analytical solutions provide exact mathematical expressions for the solution of a differential equation, while numerical solutions use computational techniques to approximate the solution through discretization and iterative calculations. Both methods have their advantages and are used depending on the complexity and nature of the differential equation being solved.

Solution of a Differential Equation Using Direction Fields

To verify the solution of a differential equation using direction fields, follow these steps:

  1. Define the Differential Equation: Start by defining the given differential equation. For example, let's consider the differential equation: \[ \frac{dy}{dx} = y - x \]

  3. Plot the Direction Field: Plot the direction field for the given differential equation. This involves drawing short line segments (arrows) at various points in the \( xy \)-plane to indicate the direction of the solution curve at each point. The slope of each segment at a point \( (x, y) \) is given by the right-hand side of the differential equation.

  5. Select Initial Conditions: Choose one or more initial conditions to determine specific solution curves. These initial conditions are of the form \( y(x_0) = y_0 \), where \( (x_0, y_0) \) is a point in the \( xy \)-plane.

  7. Trace Solution Curves: Starting from each initial condition, use the direction field to trace the solution curve. Follow the direction indicated by the arrows in the direction field to sketch the solution curve passing through the initial condition.

  9. Compare with Analytical Solution: If an analytical solution to the differential equation is known, compare the traced solution curves with the analytical solution to verify their correctness. The traced curves should match the behavior predicted by the analytical solution.