Linearization of an equation around fixed points

In summary, the process of finding the linearization of a function involves using the formula L(x) = f(a) + f'(a)(x-a) and substituting in the given fixed point values. In this specific conversation, the fixed points were found to be y=0, 1/3, and 1, and the linearization was calculated for each of these points. However, when using these fixed points, the derivative f'(a) would always be 0, making it difficult to find the linearization.
  • #1
darkspym7
10
0

Homework Statement


Find the linearization of the equation y' = y(-1+4y-3y^2) about each of the fixed points


The Attempt at a Solution


I think this is correct for finding fixed points:
Set y' = 0 = y(-1+4y+3y^2), so the fixed points are y = 0, 1/3, 1

What exactly does it mean by linearization of the equation around each of the fixed points?
 
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  • #2
linearization

the linearization of a function f about a, (linearization at x=a) is
L(x)= f(a)+f'(a)(x-a) Its pretty much like a taylor series approximation.
 
  • #3
example

find the linearization of f(x)=x^2 about x=3

L(x)=f(3)+f'(3)(x-3)
L(x)=9+6(x-3)=6x-9
 
  • #4
But using those fixed points, f'(a) would always be 0. Are those the correct fixed points?
 

Related to Linearization of an equation around fixed points

1. What is the purpose of linearization of an equation around fixed points?

The purpose of linearization is to approximate a nonlinear equation with a linear one in order to simplify analysis and calculation. This is particularly useful when studying the behavior of a system near certain fixed points.

2. How is linearization different from linear approximation?

Linearization involves finding the linear approximation of a nonlinear equation around a specific point, while linear approximation involves approximating a function with a linear one over a certain interval. Linearization is a more precise method and is used to study the behavior of a system near a fixed point.

3. What is a fixed point in the context of linearization?

A fixed point is a point at which the value of a function does not change when the function is applied to it. In other words, it is a point that remains unchanged under a certain transformation or operation.

4. Can linearization be applied to any type of equation?

No, linearization is only applicable to equations that are differentiable at the fixed point of interest. This means that the function must have a well-defined tangent line at that point.

5. How is the linearization process carried out?

The linearization process involves finding the first derivative of the function at the fixed point, which represents the slope of the tangent line. This slope is then used to create a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. This linear equation is the linearization of the original nonlinear equation around the fixed point.

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