Linearization of nonlinear non homogenous ODE

In summary, the conversation is about a non-linear non-homogeneous ODE that the speaker is trying to solve. They ask for help in reducing the equation to a Riccati equation and for any analytical methods to solve it. The other person suggests using the Bernoulli equation, but the speaker states that it is not applicable. They then discuss using the Chini equation and ask for further assistance. However, the other person points out that the original equation is separable and can be solved using partial fractions.
  • #1
mike79
9
0
Hi everybody,
could anyone help me in the linearization of the following non linear non-homogeneous ODE?

A*dy/dt+B*y^(4)=C

where A, B and C are constants. y is a function of t. is it possible to reduce this equation to a Riccati equation? do you know any analytical, approximate or not, methods to solve the equation?

thanks in advance
 
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  • #2
Just to be clear, is this the DE:

[tex]A \frac{dy}{dt} + By^4 = C[/tex]

If so, then note that you can easily express it as a http://en.wikipedia.org/wiki/Bernoulli_differential_equation" and solve it directly without having to approximate it.
 
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  • #3
the DE is right...the BERNOULLI equation is homogeneous and I actually can't tarnsform my equation in a Bernoulli one. can you suggest me how to transform it?
 
  • #4
I just had a second look at the DE and realized that there is no need to solve it as a Bernoulli DE. The original DE is separable, though the resulting integral is a little tough to integrate, but certainly doable.
 
  • #5
i have found in literature the Chini equation, which is similar to the equation I'm trying to solve. unfortunately i can't found the solution. can everyone help me, please?
 
  • #6
Defennder has already pointed out that this equation is separable:
[tex]/frac{Ady}{C- By^4}= dt[/tex]
Integrate both sides, using "partial fractions" on the left.
 

Related to Linearization of nonlinear non homogenous ODE

1. What is the purpose of linearization in solving nonlinear non homogenous ODEs?

Linearization is a mathematical technique used to approximate a non-linear system as a linear one. This allows us to apply the methods and techniques of linear systems to solve the non-linear system.

2. How is linearization performed on a nonlinear non homogenous ODE?

Linearization is typically done by taking the first-order Taylor expansion of the non-linear function at a specific point. This results in a linear function that closely approximates the non-linear one near that point.

3. Can linearization be applied to any nonlinear non homogenous ODE?

No, linearization can only be applied to certain types of non-linear systems that satisfy certain conditions. For example, the non-linear function must be differentiable and the initial conditions must be close to the point of linearization.

4. Are there any limitations to using linearization in solving nonlinear non homogenous ODEs?

Yes, linearization is an approximation technique and may not provide accurate solutions for highly non-linear systems. Additionally, the linearized system may only be valid near the point of linearization and may not accurately describe the behavior of the non-linear system as a whole.

5. Are there any alternative methods for solving nonlinear non homogenous ODEs?

Yes, there are several methods for solving non-linear systems, such as numerical methods like Euler's method or Runge-Kutta methods. However, these methods can be computationally expensive and may not always provide analytical solutions like linearization does.

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