Bernoulli Differential Equation is one of the topics that fall under calculus and differential equations. It is a nonlinear differential equation of a specific kind that can be transformed into a linear differential equation through substitution.

This article is a step-by-step guide to assisting you solve Bernoulli Differential Equations. From this method and steps, one can use it to solve other maths problems as well as problems that happen in real life.

What is Bernoulli Differential Equation?

It is known as the Bernoulli differential equation in honour of Jacob Bernoulli, a Swiss mathematician who made a significant contribution to the derivation of the equation. It is defined as:

[Tex]y’ + P(x)y = Q(x)y^n[/Tex]

Where y is the dependent variable, x is the independent variable, P(x) and Q(x) are functions of continuity and n is a real number. The standard form requires this equation to be turned into a linear form by making a substitution when n ≠ 0 and n ≠ 1.

Characteristics of Bernoulli Differential Equations

The characteristics of Bernoulli Differential Equations are as follows:

Nonlinearity: The presence of yⁿ makes the equation nonlinear, except when n=0 or n=1.

• This nonlinearity introduces complexities that make direct solutions challenging.

Transformability: It can be transformed into a linear differential equation.

• For the Bernoulli equation, there is a certain substitution that would allow the conversion to linear form for easy solving.

Dependence on n: One of the major constraints that dictate the nature of the solution is the value of n.

• Different values of n lead to different solution methods and results.

Wide Applicability: It is applied in areas such as fluids, population dynamics, and the like.

• As such its role in theoretical as well as applied area of mathematics cannot be over-emphasized.

Steps to Solve a Bernoulli Differential Equation

To solve a Bernoulli Differential Equation of the form y’ + P(x)y = Q(x)yⁿ follow these steps:

Divide through by yⁿ:

[Tex]\frac{y’}{y^n} + P(x) \frac{y}{y^n} = Q(x)[/Tex]

Simplify to:

[Tex]y^{-n}y’ + P(x)y^{1-n} = Q(x)[/Tex]

Substitute v = y¹⁻ⁿ:

This substitution helps to linearize the equation. Let v = y^{1-n}. Then, [Tex] \frac{dv}{dx} = (1-n)y^{-n} \frac{dy}{dx}[/Tex]

Rewrite the equation in terms of v:

Substituting v and its derivative into the original equation, we get:

[Tex]\frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x)[/Tex]

Solve the resulting linear differential equation:

The new equation is a linear first-order differential equation in v. Use the integrating factor method to solve it:

[Tex]\frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x)[/Tex]

The integrating factor μ(x) is:

[Tex]\mu(x) = e^{\int (1-n)P(x) \, dx}[/Tex]

Multiply through by the integrating factor:

[Tex]\mu(x)v’ + \mu(x)(1-n)P(x)v = \mu(x)(1-n)Q(x)[/Tex]

Simplify and integrate both sides:

[Tex]\frac{d}{dx} [\mu(x)v] = \mu(x)(1-n)Q(x)[/Tex]

Integrate:

[Tex]\mu(x)v = \int \mu(x)(1-n)Q(x) \, dx + C[/Tex]

Back-substitute v = y¹⁻ⁿ:

After solving for v, replace v with y¹⁻ⁿ:

[Tex]y^{1-n} = \frac{1}{\mu(x)} \left( \int \mu(x)(1-n)Q(x) \, dx + C \right)[/Tex]

Finally, solve for y:

[Tex]y = \left[ \frac{1}{\mu(x)} \left( \int \mu(x)(1-n)Q(x) \, dx + C \right) \right]^{\frac{1}{1-n}}[/Tex]

Examples of Bernoulli Differential Equations in Real Life

The examples of Bernoulli differential equations are as follows:

• Fluid Dynamics: Describes the motion of fluid under various conditions. Used in designing systems like pipelines and airfoils.

• Population Dynamics: Models population growth with carrying capacity considerations. Helps in understanding species population trends and impacts of environmental factors.

• Chemical Reactions: Represents reaction rates and concentration changes. Crucial in industrial chemistry for process optimization.

• Economics: Models economic growth and interest rates over time. Useful in financial forecasting and investment strategies.

Read More:

• Ordinary Differential Equations
• Partial Differential Equations
• Homogeneous Differential Equations

Examples of Bernoulli Differential Equations

Example 1: Solve the Bernoulli differential equation: y′ + y = xy²

Solution:

Rewrite the equation in standard form:

y′+y = xy²

Divide through by y²:

[Tex]\frac{y’}{y^2} + \frac{y}{y^2} = x[/Tex]

Simplified:

[Tex]y^{-2}y’ + y^{-1} = x[/Tex]

Substitute v = y^{-1}:

Derivative:

[Tex]\frac{dv}{dx} = -y^{-2} \frac{dy}{dx}[/Tex]

Rewrite the original equation:

[Tex]-\frac{dv}{dx} + v = x[/Tex]

Rearrange:

[Tex]\frac{dv}{dx} – v = -x[/Tex]

Find the integrating factor:

[Tex]\mu(x) = e^{\int -1 \, dx} = e^{-x}[/Tex]

Multiply through by the integrating factor:

[Tex]e^{-x} \frac{dv}{dx} – e^{-x} v = -xe^{-x}[/Tex]

Simplify:

[Tex]\frac{d}{dx} \left( e^{-x} v \right) = -xe^{-x}[/Tex]

Integrate both sides:

[Tex]e^{-x} v = \int -xe^{-x} \, dx + C[/Tex]

Using integration by parts:

[Tex]\int -xe^{-x} \, dx = -xe^{-x} – \int e^{-x} \, dx = -xe^{-x} + e^{-x}[/Tex]

Thus:

[Tex]e^{-x} v = -xe^{-x} + e^{-x} + C[/Tex]

Simplify:

[Tex]v = -x + 1 + Ce^{x}[/Tex]

Back-substitute v = y^{-1}:

[Tex]y^{-1} = -x + 1 + Ce^{x}[/Tex]

Solve for y:

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

Example 2. Solve the Bernoulli differential equation: [Tex]y′+2xy=x^3y^3[/Tex]

Solution:

Rewrite the equation in standard form:

[Tex]y′+2xy=x^3y^3[/Tex]

Divide through by y³:

[Tex]\frac{y’}{y^3} + \frac{2xy}{y^3} = x^3[/Tex]

Simplified:

[Tex]y^{-3}y’ + 2xy^{-2} = x^3[/Tex]

Substitute v = y^{-2}:

Derivative:

[Tex]\frac{dv}{dx} = -2y^{-3} \frac{dy}{dx}[/Tex]

Rewrite the original equation:

[Tex]-\frac{1}{2} \frac{dv}{dx} + 2xv = x^3[/Tex]

Rearrange:

[Tex]\frac{dv}{dx} – 4xv = -2x^3[/Tex]

Find the integrating factor:

[Tex]\mu(x) = e^{\int -4x \, dx} = e^{-2x^2}[/Tex]

Multiply through by the integrating factor:

[Tex]e^{-2x^2} \frac{dv}{dx} – 4x e^{-2x^2} v = -2x^3 e^{-2x^2}[/Tex]

Simplify:

[Tex]\frac{d}{dx} \left( e^{-2x^2} v \right) = -2x^3 e^{-2x^2}[/Tex]

Integrate both sides:

[Tex]e^{-2x^2} v = \int -2x^3 e^{-2x^2} \, dx + C[/Tex]

Using substitution u=2x², du=4xdx:

[Tex]\int -2x^3 e^{-2x^2} \, dx = \int -\frac{1}{2} u e^{-u} \, du = \frac{1}{2} (u + 1) e^{-u}[/Tex]

Thus:

[Tex]e^{-2x^2} v = \frac{1}{2} (2x^2 + 1) e^{-2x^2} + C[/Tex]

Simplify:

[Tex]v = x^2 + \frac{1}{2} + C e^{2x^2}[/Tex]

Back-substitute v = y^{-2}:

[Tex]y^{-2} = x^2 + \frac{1}{2} + C e^{2x^2}[/Tex]

Solve for y:

[Tex]y = \frac{1}{\sqrt{x^2 + \frac{1}{2} + C e^{2x^2}}}[/Tex]

Practice Problems on Bernoulli Differential Equations

P1. y’ + 3y = 2xy², Solve the Bernoulli differential equation.

P2. [Tex]y’ + (2/x)y = \sin(x) y^2[/Tex], Solve the Bernoulli differential equation.

P3. [Tex]y’ – 4y = x^2 y^3[/Tex], Solve the Bernoulli differential equation.

Conclusion

Bernoulli Differential Equation is one of the principle approaches of differential equations which gives a method to transform and solve nonlinear equations. This way with a clearly defined approach and identifying its characteristics students can solve difficult mathematical problems and apply the result in different practical sciences.

FAQs on Bernoulli Differential Equations

What are the limitations of the Bernoulli equation?

Some of the disadvantages of this Bernoulli equation include the following: Bernoulli equation applies only to a particular form of equations; also it requires a transformation to make the equation appear as a simple one, which may not be straightforward for all problems.

How to tell if a differential equation is Bernoulli?

A differential equation is Bernoulli if it can be written in the form y’ + P(x)y = Q(x)yⁿ, where P(x) and Q(x) are continuous functions and n is a real number.

Is the Bernoulli equation linear or nonlinear?

Because there is the term yⁿ when n is not equal to 0 or 1 the Bernoulli equation is nonlinear and different from linear differential equations.

What is the importance of the Bernoulli differential equation?

Indeed, the Bernoulli differential equation is used to model realistic processes for which the equations that define the growth rates depend on the current state raised to the power of some number, so its importance is justified in such areas as fluid dynamics, population growth, and much more.

What is the general solution of Bernoulli’s differential equation?

The general solution of Bernoulli’s differential equation y’ + P(x)y = Q(x)yⁿ is obtained by making the substitution v = y^{{1 – n}} and solving the resulting linear differential equation in v.