1. What is Bernoulli's Differential Equation?

The Bernoulli differential equation is a nonlinear ordinary differential equation of the form:

where \( n \) is a real number. When \( n = 0 \) or \( n = 1 \), the equation becomes linear.

2. How Does the Calculator Work?

The calculator solves Bernoulli equations using the substitution method:

This transforms the equation into a linear differential equation in \( v \), which can be solved using an integrating factor.

3. Solution Method

Step 1: Divide both sides by \( y^n \)
Step 2: Make the substitution \( v = y^{1-n} \)
Step 3: Solve the resulting linear equation for \( v \)
Step 4: Substitute back to find \( y \)

4. Using the Calculator

Tips: Enter the functions \( P(x) \) and \( Q(x) \) as mathematical expressions, and \( n \) as an integer. The dependent variable defaults to 'y' but can be changed.

5. Frequently Asked Questions (FAQ)

Q1: What values can n take?
A: n can be any real number, but the substitution method works best for integer values.

Q2: What happens when n=0 or n=1?
A: The equation becomes linear and can be solved directly with an integrating factor.

Q3: Can the calculator handle complex P(x) or Q(x) functions?
A: The calculator can process standard mathematical functions like polynomials, exponentials, and trigonometric functions.

Q4: Are initial conditions supported?
A: This version solves the general solution. For particular solutions with initial conditions, additional input would be needed.

Q5: What's the advantage of Bernoulli's equation?
A: Many physical systems can be modeled with Bernoulli equations, especially in fluid dynamics and population growth.