An autonomous differential equation is a type of differential equation where the rate of change of a variable is expressed as a function of the variable itself, and not explicitly as a function of the independent variable, usually time.

One of the key characteristics of autonomous differential equations is that their behavior is independent of the specific point in time at which you start observing the system. This feature makes them particularly useful for modeling natural phenomena where the evolution of the system depends only on its current state, such as population growth, chemical reactions, and certain mechanical systems.

What are Autonomous Differential Equations?

An autonomous differential equation is a differential equation where the independent variable does not appear in it. However, the measure of change of the dependent variable solely rests on the dependent variable. The general form of an autonomous differential equation is:

[Tex]\frac{dy}{dt} = f(y)[/Tex]

Here, y is referred to as the dependent variable, t is the independent variable, and f(y) is a function depending on the value of y only.

Note: The word ‘autonomous’ suggests that the behavior of a system depends only on the state of the system at any given time.

Examples of Autonomous Differential Equations

Some examples of the autonomous differential equations are as follows:

• Logistic Growth Model
• Simple Harmonic Oscillator
• Predator-Prey Model

Logistic Growth Model

The logistic growth model describes population growth that is self-limiting due to environmental factors. The equation is:

[Tex]\frac{dP}{dt} = rP \left(1 – \frac{P}{K}\right)[/Tex]

Here, P is the population size, r is the growth rate, and K is the carrying capacity.

Simple Harmonic Oscillator

The equation for a simple harmonic oscillator, which models systems like a mass on a spring, is:

[Tex]\frac{d^2x}{dt^2} + \omega^2 x = 0[/Tex]

Where x is the displacement, and ω is the angular frequency. Rewriting this as a system of first-order equations gives:

[Tex]\frac{dx}{dt} = v\frac{dv}{dt} = -\omega^2 x[/Tex]

Predator-Prey Model

In the Lotka-Volterra equations for predator-prey interactions, the autonomous equations are:

[Tex]\frac{dx}{dt} = \alpha x – \beta xy\frac{dy}{dt} = \delta xy – \gamma y[/Tex]

Here, x represents the prey population, y the predator population, α, β, γ, and δ are constants.

Solving Autonomous Differential Equations

Various methods can be used to solve autonomous differential equations, each providing different insights into the behavior of the solutions.

Direction Fields

Direction fields, or slope fields, graphically represent the behavior of differential equations. They help visualize solutions without solving the equation by showing the slope of the solution curve at various points. Steps to construct a direction field:

• Plot Points: Select a grid of points in the plane.
• Calculate Slopes: Compute the slope f(y) at each point.
• Draw Line Segments: At each point, draw a small line segment with the corresponding slope.
• For the equation dy/dt = y – y²: [Tex]\frac{dy}{dt} = y(1 – y)[/Tex]
• At each point (t, y), the slope is y(1 – y). Plot these slopes to create the direction field, helping to visualize the behavior of solutions.

Analytical Methods

Analytical methods involve algebraic techniques to find exact solutions. One common method is the separation of variables, where the equation is rewritten to isolate the dependent and independent variables.

Step 1: Rewrite the Equation to Separate the variables.

• [Tex]\frac{1}{f(y)} dy = dt[/Tex]

Step 2: Integrate the separated equation.

• [Tex]\int \frac{1}{f(y)} dy = \int dt[/Tex]

Step 3: Find the general solution by solving the integrated equation.

Let’s consider another example for better understanding:

For example, for the logistic equation:

[Tex]\frac{dP}{dt} = rP \left(1 – \frac{P}{K}\right)[/Tex]

Separating variables and integrating:

[Tex]\frac{dP}{P \left(1 – \frac{P}{K}\right)} = r \, dt.[/Tex]

Next, simplify the left side using partial fractions. Write:

[Tex]\frac{1}{P \left(1 – \frac{P}{K}\right)} = \frac{A}{P} + \frac{B}{1 – \frac{P}{K}}.[/Tex]

Multiplying both sides by [Tex]P \left(1 – \frac{P}{K}\right)[/Tex] gives:

[Tex]1 = A \left(1 – \frac{P}{K}\right) + BP[/Tex]

Now, equate the coefficients for P and the constant terms:

[Tex]1 = A – \frac{A}{K}P + BP.[/Tex]

This results in the system:

[Tex]A = 1 \quad \text{and} \quad B – \frac{A}{K} = 0.[/Tex]

Thus, [Tex]B = \frac{1}{K}.[/Tex]

Thus, [Tex]\frac{1}{P \left(1 – \frac{P}{K}\right)} = \frac{1}{P} + \frac{1/K}{1 – \frac{P}{K}}.[/Tex]

Rewriting this, we get

[Tex]\frac{1}{P \left(1 – \frac{P}{K}\right)} = \frac{1}{P} + \frac{1/K}{1 – P/K}.[/Tex]

So, we have:

[Tex]\int \left(\frac{1}{P} + \frac{1/K}{1 – P/K}\right) \, dP = \int r \, dt.[/Tex]

Integrate both sides:

\int \frac{1}{P} \, dP + \int \frac{1/K}{1 – P/K} \, dP = \int r \, dt.

The integrals are:

[Tex]\ln |P| – \ln |1 – \frac{P}{K}| = rt + C,[/Tex]

where C is the constant of integration.

Combine the logarithms:

[Tex]\ln \left| \frac{P}{1 – \frac{P}{K}} \right| = rt + C.[/Tex]

Exponentiate both sides to solve for P:

\left| \frac{P}{1 – \frac{P}{K}} \right| = e^{rt + C}.

Let [Tex]e^C = C_1[/Tex]:

[Tex]\frac{P}{1 – \frac{P}{K}} = C_1 e^{rt}.[/Tex]

Applications of Autonomous Differential Equations

Autonomous differential equations are applied in various fields:

• Population Dynamics: The methods, such as the logistic growth model, explain how populations increase and come to equilibrium.
• Physics: Equations that describe motion, e.g. the simple harmonic oscillator equation, are stand-alone.
• Economics: There are models that use economic growth with no regard to time such as the Solow growth model of economic growth.
• Ecology: In predator-prey phenomenon, one deals with two different species or population types.
• Engineering: Control systems often use autonomous equations to maintain system stability.

Conclusion

Optimization problems related to autonomous differential equations make up one of the key topics of mathematical modeling, as these equations give information on systems that do not depend on time. These are employed in the simplest and most complex arts of science from physics to ecology and their utility is to explain dynamic behavior. This article has provided an overview, examples, characteristics, and methods for solving autonomous differential equations, highlighting their importance and applications in various domains.

Read More,

• Differential Equation
• Seperable Differential Equation
• Homogeneous Differential Equation
• Order and Degree Of Differential Equations
• Linear Differential Equation
• First Order Differential Equation
• Ordinary Differential Equation
• Partial Differential Equation

Solved Examples of Autonomous Differential Equations

Example 1: Solve the autonomous differential equation using the separation of variables method: [Tex]\frac{dy}{dt} = y(1 – y)[/Tex]

Solution:

Rewrite the equation:

[Tex]\frac{dy}{dt} = y(1 – y)[/Tex]

Separate the variables:

[Tex]\frac{1}{y(1 – y)} dy = dt[/Tex]

Use partial fraction decomposition to rewrite the left-hand side:

[Tex]\frac{1}{y(1 – y)} = \frac{1}{y} + \frac{1}{1 – y}[/Tex]

Integrate both sides:

[Tex]\int \left( \frac{1}{y} + \frac{1}{1 – y} \right) dy = \int dt[/Tex]

Solve the integrals:

[Tex]\int \frac{1}{y} dy + \int \frac{1}{1 – y} dy = \int dt[/Tex]

Combine the logarithms and solve for P:

[Tex]\ln|y| – \ln|1 – y| = t + C[/Tex]

[Tex]\frac{y}{1 – y} = Ce^{t}[/Tex]

[Tex]y = \frac{Ce^{t}}{1 + Ce^{t}}[/Tex]

Where C is the constant of integration.

Example 2: Consider the logistic growth model given by the autonomous differential equation: [Tex]\frac{dP}{dt} = rP \left(1 – \frac{P}{K}\right) [/Tex] Solve this equation for P(t) using the separation of variables method.

Solution:

Rewrite the equation:

[Tex]\frac{dP}{dt} = rP \left(1 – \frac{P}{K}\right)[/Tex]

Separate the variables:

[Tex]\frac{1}{P \left(1 – \frac{P}{K}\right)} dP = r dt[/Tex]

Use partial fraction decomposition to rewrite the left-hand side:

[Tex]\frac{1}{P \left(1 – \frac{P}{K}\right)} = \frac{1}{P} + \frac{1/K}{1 – P/K}[/Tex]

Integrate both sides:

[Tex]\int \left( \frac{1}{P} + \frac{1}{K – P} \right) dP = \int r dt[/Tex]

Solve the integrals:

[Tex]\int \frac{1}{P} dP + \int \frac{1}{K – P} dP = \int r dt[/Tex]

Combine the logarithms and solve for P:

[Tex]\ln \left| \frac{P}{K – P} \right| = rt + C[/Tex]

[Tex]\frac{P}{K – P} = Ce^{rt}[/Tex]

[Tex]P = \frac{Ce^{rt} K}{1 + Ce^{rt}}[/Tex]

Where C is the constant of integration.

Solve for the initial condition P(0)=P₀:

[Tex]P(0) = \frac{C K}{1 + C} = P_0[/Tex]

[Tex]C = \frac{P_0}{K – P_0}[/Tex]

Substitute C back into the solution:

[Tex]P = \frac{\frac{P_0}{K – P_0} K e^{rt}}{1 + \frac{P_0}{K – P_0} e^{rt}}[/Tex]

Simplify the expression:

[Tex]P = \frac{P_0 K e^{rt}}{K – P_0 + P_0 e^{rt}}[/Tex]

Final solution:

[Tex]P(t) = \frac{K P_0 e^{rt}}{K – P_0 + P_0 e^{rt}}[/Tex]

Practice Problems on Autonomous Differential Equations

Problem 1: Solve the differential equation:

[Tex]\frac{dy}{dt} = y^2 – y – 6[/Tex]

Problem 2: Solve the autonomous differential equation with a damping term:

[Tex]\frac{dy}{dt} = -y^3 + y[/Tex]

Problem 3: Solve the following autonomous differential equation that models a chemical reaction:

[Tex]\frac{dy}{dt} = y(1 – y^2)[/Tex]

FAQs on Autonomous Differential Equations

Define Autonomous Differential Equations.

An autonomous differential equation is a differential equation in which the independent variable, typically time t, does not explicitly appear in the equation.

How to determine if a differential equation is autonomous?

A differential equation is autonomous if it can be written in the form dy/dt = f(y), meaning the independent variable t does not explicitly appear in the function f.

Are all autonomous differential equations separable?

Yes, all autonomous differential equations are separable. They can be expressed in the form dy/f(y) = dt and then integrated on both sides.

What is the difference between autonomous and homogeneous differential equations?

An autonomous differential equation does not explicitly depend on the independent variable, while a homogeneous differential equation’s terms can be grouped into functions of the same degree when scaled.

What are the slope fields for autonomous differential equations?

The slope fields for autonomous differential equations have slopes that depend only on the dependent variable y and not on the independent variable t. This results in horizontal translation symmetry in the slope field.