Section 19.4
Bernoulli Equations
A clever substitution turns a scary non-linear equation into a friendly linear one.
1
Introduction
A Bernoulli Equation is a First Order DE of the form:
If or , it's already linear. But for any other , that term ruins the linearity. We fix this by substituting.
2
The Method
Recipe
- Divide: Divide by to get .
- Substitute: Let . This implies .
- Transform: The equation becomes .
- Solve: Solve this new Linear DE for .
- Back-Sub: Replace with to get the final answer.
3
Visual: Logistic Growth
Interactive: The most famous Bernoulli
The Logistic Equation is simpler to solve as a Bernoulli equation than by Partial Fractions.
4
Worked Examples
Example 1: Basic Application
Solve .
1. Identify n:
. Substitute .
2. Transform:
Divide by : .
Since , we have .
Standard Linear Form: .
3. Solve Linear:
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4. Back-Sub:
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Example 2: Variable Coefficients
Solve , .
1. Identify n:
. .
2. Transform:
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3. Integrating Factor:
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4. Integrate:
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5. Final:
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Example 3: Cubic Bernoulli
Solve for .
1. Standard Form:
. Here .
2. Substitute:
. .
Divide DE by : .
Multiply by -2: .
.
3. Solve Linear:
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4. Back-Sub:
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5
Practice Quiz
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