Question

Solving a Logistic Differential Equation In Exercises \(55-58\) , find the logistic equation that passes through the given point. $$ \frac{d y}{d t}=2.8 y\left(1-\frac{y}{10}\right), \quad(0,7) $$

Short answer

The logistic equation that passes through the point (0,7) is \(y = \frac{1}{ce^{-2.8t}}\), where c is determined by the point (0,7).

Step-by-step solution

- Identify the key components

The differential equation is given as \(\frac{dy}{dt} = 2.8 \cdot y(1-\frac{y}{10})\). Here, \(r = 2.8\) and \(K = 10\). The point through which the logistic equation passes is (0,7).

- Apply the Bernoulli's substitution

Bernoulli's equation is \(y' + p(x)y = q(x)y^n\). If we rearrange our equation we get \(\frac{dy}{dt} + 2.8 \frac{y^2}{10} - 2.8 y = 0\). Therefore \(p(x) = 2.8\), \(q(x) = -2.8\), and \(n = 2\). Bernoulli's substitution is \(v = y^{1-n}\) or \(v = y^{-1}\).

- Transform into linear differential equation

Transforming our logistic differential equation into linear form using \(v = y^{-1}\), we get an equation for v as \(\frac{dv}{dt} = -2.8 v\).

- Solve the linear differential equation

Use the integrating factor to solve this linear differential equation. We get \(v = ce^{-2.8t}\).

- Apply reverse substitution

We put \(v = y^{-1}\), and get \(\frac{1}{y} = ce^{-2.8t}\), or \(y = \frac{1}{ce^{-2.8t}}\).

- Find constant of integration

Substitute the given point (0,7) into the equation to find the constant c. Then give the final logistic equation.

Key concepts

Bernoulli's Equation

Before diving into the complex world of logistic differential equations, it's essential to understand Bernoulli's equation. This form of differential equation is characterized by an equation like \(y' + p(x)y = q(x)y^n\), where \(n\) is a real number, and \(p(x)\) and \(q(x)\) are functions of \(x\). Bernoulli's equation becomes a linear differential equation when \(n=0\) or \(n=1\).

In the exercise, we encounter a logistic differential equation that can be transformed into a Bernoulli's equation by identifying an appropriate substitution. By recognizing that the given equation resembles the standard form of Bernoulli's equation with \(n=2\), a substitution is made, facilitating the simplification to a linear form. This crucial first step towards solving a logistic equation cannot be overstated, as it sets the foundation for the remaining solving process.

Understanding how the parameters such as \(r\) and \(K\) in the logistic differential equation relate to \(p(x)\) and \(q(x)\) in Bernoulli's equation is vital for correctly setting up and solving the problem.

Integrating Factor

The concept of an integrating factor is a powerful tool used for solving linear differential equations. Specifically, when we have an equation in the form \(\frac{dv}{dt} + P(t) \times v = Q(t)\), an integrating factor \(\mu(t)\) can be obtained through the formula \(\mu(t) = e^{\int P(t) \text{ d}t}\).

When multiplying the entire differential equation by this integrating factor, the left side of the equation becomes the derivative of the product of the factor and the function \(v\), which considerably simplifies the equation. After this, integration on both sides of the equation is much more straightforward, leading us closer to the solution. In our exercise, the integrating factor helped transform the equation involving \(v\) to a form that can be integrated directly, allowing us to find \(v\) in terms of \(t\) and the constant \(c\). This is a critical step in the process of finding the solution to the differential equation.

Linear Differential Equation

A linear differential equation is an equation that involves unknown functions with their derivatives, and every term is either a constant or the product of a constant and a first derivative of the function. After applying Bernoulli's substitution in the logistic differential equation, we attain a linear differential equation of the form \(\frac{dv}{dt} = -2.8v\).

This simplification to a linear form makes it possible to employ standard techniques for solving linear equations, such as finding an integrating factor and integrating both sides of the equation. It's essential to acknowledge the importance of transforming complex equations into linear differential equations. It simplifies the process significantly and leverages well-established methods for finding the solutions, ensuring the problem becomes manageable and leads to a successful outcome.

Constant of Integration

In the context of differential equations, when integrating, we obtain an arbitrary constant, known as the constant of integration. This constant represents the family of all possible solutions to the differential equation because integration is, in essence, the inverse operation of differentiation, and it undoes the precise changes implemented by the derivative.

For the given logistic differential equation, once the integrating process is complete, we incorporate the constant of integration \(c\) to express the general solution. To determine the specific solution that corresponds to the initial condition given by the point (0,7), we substitute these values into our general solution. By doing so, we can solve for the particular value of \(c\) that satisfies the condition, thereby pinpointing the unique solution to the problem amidst the infinite family of potential solutions.