Bernoulli Equation Calculator Differential Equations

Bernoulli Differential Equation:

\[ y' + P(x)y = Q(x)y^n \]

Solution:

\[ y = \left[ (1-n) \int Q e^{(1-n) \int P dx} dx + C \right]^{1/(1-n)} \]

Solution (y):

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1. What is the Bernoulli Differential Equation?

The Bernoulli differential equation is a nonlinear first-order differential equation of the form y' + P(x)y = Q(x)y^n. It can be transformed into a linear differential equation and solved using an integrating factor.

2. How Does the Calculator Work?

The calculator uses the Bernoulli equation solution formula:

\[ y = \left[ (1-n) \int Q e^{(1-n) \int P dx} dx + C \right]^{1/(1-n)} \]

Where:

\( n \) — The exponent in the original equation (integer)
\( P(x) \) — The coefficient function of y
\( Q(x) \) — The function on the right side
\( C \) — Constant of integration

Explanation: The solution involves computing two integrals - first the integral of P(x), then using that to compute the integral of Q(x) multiplied by an exponential factor.

3. Importance of Bernoulli Equations

Details: Bernoulli equations appear in various physical applications including population dynamics, fluid mechanics, and thermodynamics. They provide a model for nonlinear phenomena that can be transformed into linear problems.

4. Using the Calculator

Tips: Enter the exponent n (must be an integer), the functions P(x) and Q(x) as mathematical expressions, and the constant C. The calculator will display the general solution structure.

5. Frequently Asked Questions (FAQ)

Q1: What values can n take?
A: n can be any integer except 1 (when n=1, the equation becomes linear and is solved differently).

Q2: How do I enter functions?
A: Use standard mathematical notation like "2x", "sin(x)", "e^x", or "x^2".

Q3: What if I get an error?
A: Check that n is an integer and your functions are properly formatted. Some complex functions may require exact syntax.

Q4: Does this solve specific IVPs?
A: This provides the general solution. For an initial value problem, you would substitute initial conditions to solve for C.

Q5: What are common applications?
A: Bernoulli equations model phenomena like population growth with competition, fluid flow with friction, and chemical reaction kinetics.