Can Anyone Help with Linearization of Equations Around x=theta=0?

[e135193]

Is there anyone that can help me about linearization of the equations around x=teta=0 ? It is a bit urgent. I am waiting answer from the calculus experts.

 

[tiny-tim]

Welcome to PF!

Is there anyone that can help me about linearization of the equations around x=teta=0 ? It is a bit urgent. I am waiting answer from the calculus experts.

Hi e135193! Welcome to PF!

hmm … (c cosθ, c sinθ) is the point at angle θ on the circle of radius c round the origin.

r1 is the distance from that point, and r2 is the distance from the opposite point, (-c cosθ, -c sinθ).

but I don't understand what you mean by "linearization of the equations around x = θ = 0"

 

[e135193]

As you see equations are non-linear. I need equations like in the picture below. I heard that it can be made by using taylor series.

 

[tiny-tim]

As you see equations are non-linear. I need equations like in the picture below. I heard that it can be made by using taylor series.

Hi e135193!

(btw, anyone who contributes to a thread gets an email whenever someone else replies, so there's no need to use PMs. )

This may help: √(1 + x) = (1 + x/2 + …); sinx = x + … ; cosx = 1 - x²/2 + …

But what I don't understand is that you have three variables, x y and θ, for a 2-dimensional situation.

 

[e135193]

Thanks for your interest to help me.

I'm sorry about the insufficient information. The "y" and "c" terms are constants.
Then, there are only 2 variables, x and θ.

Can you help me further from that because I am not good at mathematics.

 

[e135193]

Revised Question...

This is the clear definiton of my problem.. Does Anyone Know the solution ?

 

[HallsofIvy]

I once walked into a test on differential equations that asked for the "linearization" of a non-linear de and I had no idea what "linearization" meant! Turns out it's easy.

You have
$$r_1= \sqrt{(c cos(\theta)- x)^2+ (y- c sin(\theta)^2}$$
$$r_2= \sqrt{(c cos(\theta)+ x)^2+ (y+ c sin(\theta)^2}$$
and you say that c and y are constant. So you want linear functions for r₁ and r₂ in terms of x and $\theta$. You don't really need the full "Taylor's series", you just need the tangent line equations (which are the order one Taylor's Polynomials).

In general, if r= f(x, $\theta$), then the "tangent plane" approximation, about x= 0, $\theta= 0$, is given by
$$r= f(0,0)+ \frac{\partial f}{\partial x}(0,0) x+ \frac{\partial f}{\partial \theta}(0,0)\theta$$
Find the partial derivatives of your functions and plug into that.

 

[e135193]

thank you so much for your effort. I'm going to try to do but its not going to be easy for me. But the point I smash is that I can not plug that functions into the expression you define above. Can you please a bit help )

 

[tiny-tim]

thank you so much for your effort. I'm going to try to do but its not going to be easy for me. But the point I smash is that I can not plug that functions into the expression you define above. Can you please a bit help )

Hi e135193!

ok, y and c are constants.

So r is a function of x and θ.

What is the derivative ∂r/∂x? It's the rate at which r increases if you increase x (but leave θ fixed).

So if you only go a very small distance (in x), then r depends linearly on x, and the factor is ∂r/∂x. Similarly for ∂r/∂θ.

Work out ∂r/∂x and ∂r/∂θ, and then find their values at (x=0,θ=0).

 

[e135193]

hi ,tiny-tim.

Thanks for your effort again but I said that I was not good at mathematics.


I can not take the partial derivative :) of that function.

could you please explain a bit further.


kind regards.

 

[tiny-tim]

hi ,tiny-tim.

Thanks for your effort again but I said that I was not good at mathematics.

I can not take the partial derivative :) of that function.

could you please explain a bit further.


kind regards.

ok, let's start with an example: ∂/∂x of √(x² + px + q) … can you do that?

(if not, just d/dx of √(x² + px + q)? )

 

[e135193]

if I said no, would you angry of me ?

 

[tiny-tim]

ok, just try d/dx of x² + px + q.

You can do that, can't you?

Do you then know how to use substitution to get d/dx of √(x² + px + q)?

hmm … got to go to bed now … :zzz:

 

[e135193]

hi tiny-tim

I'm begging you please. It's very important for me. I have to finish this before the sun rises .

Again I'm saying I'm not good at mathematics.


There is someone who waits for answer.

 

[tiny-tim]

Hi e135193!

I have to finish this before the sun rises

But I don't know your time-zone!

And what part of …

hmm … got to go to bed now … :zzz:

… did you not understand?

Look, if you really can't do d/dx of x² + px + q, then you'd better ask your teacher/TA for special help.

That sort of d/dx is something you should be able to do in your head by now, like seven times eight equals fifty-six.

But if you can do it, show us what it is, and we'll help you through the next steps.

 

[HallsofIvy]

If you do not know how to differentiate at all (which is what you seem to be saying) then you should not trying to do a problem like this. This kind of problem is typically given in a calculus course about a year after learning things like how to differentiate x²+ px+ q.

 

[e135193]

Hi tiny-tim,


Finally, I have calculated equations but I don't have any idea about whether they are correct or nor. Atleast could you check it please ?

 

[tiny-tim]

Hi e135193!

I'm sorry, but i have no idea how you got that …

are they supposed to be ∂r1/∂x and ∂r2/∂x, or what?

and where did those √(c² + y²) some from?

We'd better start with:

what is d/dx of √(x² + px + q)?

 

[e135193]

equations I found is the solution of my equations not the one you suggested me to solve

The equation I found is that the linearized one of the non-linear equations.

I just asked that is that true ? could you check it ?

 

[tiny-tim]

The equation I found is that the linearized one of the non-linear equations.

oh … now I understand.

Your -cx/√(c² + y²) and your + √(c² + y²) are correct,

but I've no idea how you got the cos(c) and the sin(y) in the middle (presumably from your ∂/∂θ).

Do the ∂/∂θ again, and this time show your working, so we can see where you're going wrong.

 

[e135193]

I used Mathematica for that partial derivative.

you say it is wrong. Then show the corrcet solution.

please I have to do this otherwise I will fail and I lose my scholarship.

 

[tiny-tim]

Hi e135193!

I used Mathematica for that partial derivative.

Words almost fail me!

You clearly have no idea how to do elementary calculus.

you say it is wrong. Then show the corrcet solution.

No-one on this forum will do your work for you.

I've tried to talk you through this problem, but you haven't cooperated, and it turns out you can't even use Mathematica properly, or even recognise when it churns out nonsense.

please I have to do this otherwise I will fail and I lose my scholarship.

If you can't do elementary calculus in your head, are you even doing the right course for you?

e135193 , I think you'd better come clean and start a new thread in the Academic & Career Guidance sub-forum, and explain your present academic situation and ask for advice.

There are plenty of people on this forum who have experience of your situation, and who can advise you, if you ask.