Solve Bernoulli Equation x'=b(t)x+c(t)x^n when n<0

In summary, the Bernoulli equation is a first-order ordinary differential equation used to model various physical phenomena. It has the form of x' = b(t)x + c(t)x^n, where x is the dependent variable, t is the independent variable, and b(t) and c(t) are functions of t. The n<0 condition indicates a non-linear equation that cannot be solved using standard techniques. However, it can be solved using the substitution method by transforming it into a linear differential equation. The Bernoulli equation has many applications in physics, chemistry, biology, and economics, but it has limitations and may not accurately represent complex systems.
  • #1
The|M|onster
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How do I solve a Bernoulli equation of the form x' = b(t)x + c(t)x^n when n < 0?
 
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  • #2
The|M|onster said:
How do I solve a Bernoulli equation of the form x' = b(t)x + c(t)x^n when n < 0?


[tex]\frac{dx}{dt}-b(t)x=c(t)x^{n}[/tex] A standard substitution is [tex]v=x^{1-n}[/tex]
 
  • #3
Thank you very much. it worked perfectly.
 

1. What is the Bernoulli equation?

The Bernoulli equation is a type of first-order ordinary differential equation that is used to model a wide variety of physical phenomena, such as fluid flow, chemical reactions, and population growth. It is in the form of x' = b(t)x + c(t)x^n, where x is the dependent variable, t is the independent variable, and b(t) and c(t) are functions of t.

2. What does the n<0 condition mean in the Bernoulli equation?

The n<0 condition in the Bernoulli equation indicates that the dependent variable x is raised to a negative power. This makes the equation non-linear, and it cannot be solved using standard techniques for linear differential equations.

3. How do you solve the Bernoulli equation when n<0?

When n<0, the Bernoulli equation can be solved using the substitution method. By substituting y = x^(1-n), the equation can be transformed into a linear differential equation, which can then be solved using standard techniques.

4. What are some applications of the Bernoulli equation?

The Bernoulli equation has many applications in physics, chemistry, and biology. It is commonly used to model fluid flow in pipes and channels, chemical reactions, and population growth. It is also used in economics to model the relationship between supply and demand.

5. Are there any limitations to the Bernoulli equation?

Yes, the Bernoulli equation has some limitations. It can only be used to model systems that exhibit exponential growth or decay. Additionally, it assumes that the coefficients b(t) and c(t) are constant, which may not always be the case in real-world applications. It is also important to note that the Bernoulli equation is a simplified model and may not accurately represent complex systems.

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