Do you understand what linearization means?

[hihi]

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

 

[HallsofIvy]

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.

 

[hihi]

x'' means diferentiation twice.however, i think it should be solve by using taylor series.Am i correct??But i still can do it...

 

[HallsofIvy]

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.

 

[hihi]

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...

 

[Ben Niehoff]

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.

 

[hihi]

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

 

[hihi]

pls help...thanx

 

[HallsofIvy]

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?