Do you understand what linearization means?

In summary: I really don't get why people do this.1. Homework Statement Linearization4x'' + 3 cos(x-y)y'' +2 y^2 sin(x-y) +3g sin (x) = 03 cos(x-y) y'' + 2y'' - 2y' ^2 sin(x-y) +g sin(y) =02. Homework Equations initial condition x(0) = y(0) = 03. The Attempt at a Solution the answer for this question is 4x'' + 3y'' + 3g x = 0 and 3x''
  • #1
hihi
6
0
do you understand what "linearization" means?

Homework Statement



Linearization
4x'' + 3 cos(x-y)y'' +2 y^2 sin(x-y) +3g sin (x) = 0

Homework Equations



initial condition x(0) = y(0) = 0


The Attempt at a Solution



pls tell me the relevant steps to solve this problem.thanx
 
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  • #2
Do you understand what "linearization" means? (I ask because I finally realized how simple it was the day after a test!)

Just replace any non-linear functions of the dependent variable by linear approximations.

It would help, of course, to separate dependence on the different variables (am I correct that both x and y are dependent variables, depending on a third variable and the derivatives are with respect to that third variable?): using the trig sum formulas cos(x- y)= cos(x)cos(y)+ sin(x)sin(y) and sin(x-y)= sin(x)cos(y)- cos(x)sin(y). Since you will want to approximate around x= y= 0, the linear approximation to sin(x) is x near x= 0 and to cos(x) is 1 around x= 0. (You can see that by looking at their Taylor's series.) Of coure, the only "linear" approximation to y2 or xy around y= 0 is 0.
 
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  • #3
x'' means diferentiation twice.however, i think it should be solve by using taylor series.Am i correct??But i still can do it...
 
  • #4
My only point about Taylor's series is that by dropping all the terms of degree 2 or above, you get the "linearization".
sin(x)= x- x3/3!+ ... and so the "linearization" of sin(x), around x= 0, is x.
cos(x)= 1- x2/2!+... so the "linearization" of cos(x), around 0, is 1.
 
  • #5
sorry,there is another equation given which is 3 cos(x-y) y'' + 2y'' - 2y' ^2 sin(x-y) +g sin(y) =0 .

but,the answer for this question is 4x'' + 3y'' + 3g x = 0 and 3x'' + 2y'' + gy = 0.

pls help...
 
  • #6
If you can't be bothered to type in complete sentences, and clearly state your point of confusion, why should somebody take the time to explain anything to you?

I really don't get why people do this.
 
  • #7
1. Homework Statement

Linearization

4x'' + 3 cos(x-y)y'' +2 y^2 sin(x-y) +3g sin (x) = 0

3 cos(x-y) y'' + 2y'' - 2y' ^2 sin(x-y) +g sin(y) =0
2. Homework Equations

initial condition x(0) = y(0) = 03. The Attempt at a Solution

the answer for this question is 4x'' + 3y'' + 3g x = 0 and 3x'' + 2y'' + gy = 0.

pls tell me the relevant steps to solve this problem.thanx
 
  • #8
pls help...thanx
 
  • #9
hihi said:
sorry,there is another equation given which is 3 cos(x-y) y'' + 2y'' - 2y' ^2 sin(x-y) +g sin(y) =0 .

but,the answer for this question is 4x'' + 3y'' + 3g x = 0 and 3x'' + 2y'' + gy = 0.

pls help...
?? Are you saying that you started with two equations:
4x'' + 3 cos(x-y)y'' +2 y^2 sin(x-y) +3g sin (x) = 0
and
3 cos(x-y) y'' + 2y'' - 2y' ^2 sin(x-y) +g sin(y) =0 .

And it didn't occur to you to tell us that?
 

FAQ: Do you understand what linearization means?

1. What is linearization?

Linearization is a mathematical process of approximating a nonlinear function with a linear function. It involves finding the first-order derivative of the function and evaluating it at a specific point.

2. Why is linearization important in science?

Linearization is important in science because many natural phenomena can be modeled using nonlinear functions. By linearizing these functions, we can make them easier to analyze and understand, and also make predictions and solve problems more accurately.

3. What is the purpose of linearization in data analysis?

In data analysis, linearization is used to transform nonlinear data into a linear form. This makes it easier to apply statistical methods and analyze the data, as linear relationships are more easily interpreted and modeled. It also helps to identify trends and patterns in the data.

4. Can you give an example of linearization in real life?

One example of linearization in real life is the use of logarithmic scales. In certain situations, data may follow a logarithmic relationship, which can be linearized by taking the logarithm of the data. This allows for easier analysis and interpretation, and is commonly used in fields such as economics and biology.

5. How is linearization different from linear regression?

Linearization and linear regression are two different techniques used for modeling data. Linearization involves transforming nonlinear data into a linear form, while linear regression is a statistical method used to find the best fitting line for a given set of data. Linearization is often used as a preliminary step before applying linear regression to data.

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