Global solution to inhomogeneous Bernoulli ODE

In summary, a global solution to an inhomogeneous Bernoulli ODE refers to a solution that is valid over the entire domain of the independent variable. This is different from a local solution, which is only valid within a specific interval or range. An inhomogeneous Bernoulli ODE is a type of ODE that is useful in modeling various physical and biological phenomena. The global solution to this type of ODE has applications in fields such as science, engineering, and economics. It can be obtained using various mathematical techniques, such as integrating factors and power series solutions, depending on the initial conditions given.
  • #1
doobly
2
0
Hi everyone,

I have an inhomogeneous Bernoulli type ODE given by

[itex] u'(t) = \kappa u(t) + \ell(t) u^{\gamma}(t) + v(t),\ \ \ u(T)=b>0,...(1) [/itex]

where [itex] t\in[0,T],\ \ \gamma\in (0,1) [/itex].

My concern is that how to prove the existence and uniqueness of the solution u(t) for all [itex]t\in [0,T] .[/itex] Thanks very much.
 
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  • #2
As long as l(t) and v(t) are "Lipschitz" ("differentiable" is sufficient but not necessary) on [0, 1], that follows from the elementary "existance and uniqueness" theorem for intial value prolems of the for equations of the form y'= f(t, y).
 

FAQ: Global solution to inhomogeneous Bernoulli ODE

1. What is a "Global Solution" to an inhomogeneous Bernoulli ODE?

A global solution to an inhomogeneous Bernoulli ODE refers to a solution that is valid over the entire domain of the independent variable. In other words, it is a solution that holds true for all possible values of the independent variable, not just a specific range or interval.

2. How is a "Global Solution" to an inhomogeneous Bernoulli ODE different from a "Local Solution"?

A local solution to an inhomogeneous Bernoulli ODE is a solution that is valid only within a specific interval or range of the independent variable. It may not hold true for values outside of this range. A global solution, on the other hand, is valid for all possible values of the independent variable.

3. What is an inhomogeneous Bernoulli ODE?

An inhomogeneous Bernoulli ODE is a type of ordinary differential equation (ODE) that can be written in the form of y' + p(x)y = q(x)y^n + g(x), where y is the dependent variable, x is the independent variable, p(x), q(x), and g(x) are known functions of x, and n is a constant. This type of ODE is useful in modeling various physical and biological phenomena.

4. What are some applications of the "Global Solution" to inhomogeneous Bernoulli ODEs?

The "Global Solution" to inhomogeneous Bernoulli ODEs has many applications in various fields of science and engineering. It can be used to model population growth, chemical reactions, and heat transfer processes. It is also commonly used in economic models, epidemiology, and ecology.

5. How can a "Global Solution" to an inhomogeneous Bernoulli ODE be obtained?

A "Global Solution" to an inhomogeneous Bernoulli ODE can be obtained using various mathematical techniques, such as the method of integrating factors, substitution methods, and power series solutions. The choice of method depends on the specific form of the ODE and the initial conditions given. In some cases, numerical methods may also be used to approximate the global solution.

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