Inhomogenous NON-linear differential equation

However, it is not clear if this substitution would lead to an exact solution.In summary, the conversation is about solving a non-linear ODE in the form of a Bernoulli equation. The speaker is unsure if there exists an exact solution and wonders if a substitution could help solve it. The suggested substitution is x'' = x'(dx'/dx), but it is uncertain if it would lead to an exact solution.
  • #1
Thoughtknot
1
0
I'm having some trouble solving an equation that is similar to a Bernoulli equation. It is of the form

\begin{equation}
\ddot{x}+f(x)\dot{x}^2 = g(x)
\end{equation}

Where x is a function of time, perhaps. I feel moderately certain that there should exist an exact solution, but I've so far been unable to find it, and I have not run into any great amount of non-linear ODEs before.

Does anyone have any idea if it can be solved? Could it be solved by some clever substitution?
 
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  • #2
The substitution x'' = x'(dx'/dx) would reduce this to a Bernoulli differential equation.
 

FAQ: Inhomogenous NON-linear differential equation

What is an inhomogeneous non-linear differential equation?

An inhomogeneous non-linear differential equation is a type of mathematical equation that involves a function and its derivatives. It is called "inhomogeneous" because the function is not equal to zero, and "non-linear" because the function and its derivatives are raised to powers other than 1.

What are some real-life applications of inhomogeneous non-linear differential equations?

Inhomogeneous non-linear differential equations are used in many areas of science and engineering, such as physics, biology, economics, and chemistry. They can be used to model complex systems and phenomena, such as population growth, chemical reactions, and electrical circuits.

How do you solve an inhomogeneous non-linear differential equation?

Solving an inhomogeneous non-linear differential equation can be a complex process and often requires advanced mathematical techniques. It typically involves finding the general solution using integration or other methods, and then applying boundary conditions or initial conditions to find a specific solution.

What are some common challenges when working with inhomogeneous non-linear differential equations?

One of the main challenges when working with inhomogeneous non-linear differential equations is that they do not have a general method for solving them. Each equation must be approached individually, and the methods used will depend on the specific form of the equation and the initial or boundary conditions given.

How are inhomogeneous non-linear differential equations related to other types of differential equations?

Inhomogeneous non-linear differential equations are a specific type of differential equation that falls under the broader category of non-linear differential equations. They are also related to homogeneous non-linear differential equations, which have a function equal to zero, and linear differential equations, which have derivatives raised to the first power.

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