bernoulli equation in differential equation

3 min read 19-10-2024

bernoulli equation in differential equation

Demystifying the Bernoulli Equation: A Comprehensive Guide for Differential Equations

The Bernoulli equation is a fundamental concept in differential equations, allowing us to solve a specific type of non-linear equation by transforming it into a linear form. This article will delve into the intricacies of the Bernoulli equation, providing you with a comprehensive understanding of its application and how to solve it.

What is a Bernoulli Equation?

A Bernoulli equation is a first-order differential equation of the form:

dy/dx + p(x)y = q(x)y^n

where p(x) and q(x) are continuous functions of x, and n is a real number not equal to 0 or 1.

Key Characteristics:

Non-linear: The presence of the y^n term makes the equation non-linear.
First-order: The equation involves the first derivative of the dependent variable y.
Specific Form: The equation follows a specific pattern with the y^n term multiplied by q(x).

Why is the Bernoulli Equation Important?

The significance of the Bernoulli equation lies in its ability to be transformed into a linear equation, which can be solved using established methods. This transformation is achieved by a clever substitution technique, making it a powerful tool for solving a specific class of non-linear differential equations.

The Solution Strategy: Transformation and Integration

1. Substitution:

The key to solving a Bernoulli equation is to introduce a new dependent variable, u, defined as:

u = y^(1-n)

2. Differentiation:

Differentiate both sides of the substitution equation with respect to x:

du/dx = (1-n)y^(-n) dy/dx

3. Rearranging:

Rearrange the equation to express dy/dx in terms of u and du/dx:

dy/dx = (y^n / (1-n)) du/dx

4. Substituting into the Original Equation:

Substitute the expressions for dy/dx and y^n (which can be expressed as u^(1/(1-n))) into the original Bernoulli equation:

(y^n / (1-n)) du/dx + p(x)y = q(x)y^n

Simplifying the equation, we obtain:

du/dx + (1-n)p(x)u = (1-n)q(x)

5. Solving the Linear Equation:

This equation is now a linear first-order differential equation in terms of u. This equation can be solved using standard techniques such as the integrating factor method:

Find the Integrating Factor: I(x) = exp(∫(1-n)p(x) dx)
Multiply both sides by the Integrating Factor: I(x) du/dx + (1-n)p(x)I(x)u = (1-n)q(x)I(x)
Simplify: This step leads to the derivative of a product: d/dx (I(x)u) = (1-n)q(x)I(x)
Integrate both sides: ∫d(I(x)u) = ∫(1-n)q(x)I(x) dx
Solve for u: u = (1/I(x)) ∫(1-n)q(x)I(x) dx + C
Substitute back for y: Finally, substitute y = u^(1/(1-n)) to obtain the solution for y.

Example:

Consider the Bernoulli equation:

dy/dx + 2y = y^3

Following the steps outlined above:

Substitution: u = y^(1-3) = y^(-2)
Differentiation: du/dx = -2y^(-3) dy/dx
Rearranging: dy/dx = (-y^3/2) du/dx
Substituting: (-y^3/2) du/dx + 2y = y^3
Simplifying: du/dx - 4u = -2
Integrating Factor: I(x) = exp(∫-4 dx) = e^(-4x)
Solving: u = e^(4x) (∫ -2e^(-4x) dx + C) = e^(4x) (1/2e^(-4x) + C) = 1/2 + Ce^(4x)
Substituting back for y: y = u^(-1/2) = 1/√(1/2 + Ce^(4x))

Therefore, the general solution of the Bernoulli equation is y = 1/√(1/2 + Ce^(4x)).

Conclusion

The Bernoulli equation is a valuable tool for solving a specific type of non-linear differential equation. By utilizing a clever substitution technique, it transforms the non-linear equation into a solvable linear form, enabling us to find solutions that might otherwise be inaccessible. This article has explored the key concepts, solution strategies, and an example application of the Bernoulli equation, providing you with a solid foundation for understanding and solving this important class of differential equations.

Further Resources:

"Differential Equations" by Dennis G. Zill and Michael R. Cullen: This textbook provides a comprehensive overview of differential equations, including detailed explanations of the Bernoulli equation.
Khan Academy: Offers video lectures and practice problems on differential equations, including the Bernoulli equation.

Note: This article is based on information from various resources, including Stack Overflow and other online forums. The specific contributions of individual authors are difficult to pinpoint due to the collaborative nature of these platforms. This article aims to synthesize and present this information in a clear and informative manner.