# Why is this differential equation non-linear?

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• PainterGuy
In summary, a rocket's thrust is equal to the exhaust velocity times the rate of change of mass, which is known as the mass flow rate of exhaust. This relationship is represented by the equation {T}={v}\frac{dm}{dt}. The differential equation for this system is non-linear due to the presence of the non-linear term u=\overset{\cdot }{m}. However, it is common practice to approximate non-linear systems with linear models. This can be done by using matrix equations or through perturbation theory, but it is important to ensure the accuracy and applicability of the model. Further study and research, such as the lectures by Steven Strogatz, can provide a deeper understanding of linearization in modern control
PainterGuy
Hi,

Could you please have a look on the attachment?

Question 1:
Why is this differential equation non-linear? Is it $u=\overset{\cdot }{m}$ which makes it non-linear?

I think one can consider $x_{3}$ , k, and g to be constants. If it is really $u=\overset{\cdot }{m}$ which makes it non-linear then I don't think it's possible to make it linear. Could you please correct me?

Question 2:
What is there in "These expressions" which shows that the system is linear? Is it "0 x u"?

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Because it contains a non-linear term regardless of how you write it.

PainterGuy
Mayhem said:
Because it contains a non-linear term regardless of how you write it.

Thank you!

Is that non-linear term $u=\overset{\cdot }{m}$?

What is there in that equation system which shows that the system is not linear?

Thanks a lot for your time and help!

PainterGuy said:
Thank you!

Is that non-linear term $u=\overset{\cdot }{m}$?

What is there in that equation system which shows that the system is not linear?

Thanks a lot for your time and help!
Since ##u = \dot{m}## and ##x_3 = m## it means that ##\frac{k}{x_3}u = \frac{k}{m}\dot{m}##. Usually, products of the same function (like ##y^n##, where ##n## is some number) and products of functions/dervative of functions (like ##y'y''## or ##yy'''## and similar) is a quick way to qualitatively check if a differential equation is non-linear. I might be wrong though. My only real training in differential equations consist of a mandatory UG course on analysis.

PainterGuy
In the matrix equation you have for ##x_1,x_2,x_3##, the equation would be linear if the matrix was full of constants. But it has an ##x_3## term in it.

DaveE and PainterGuy
You said you thought that x3 = m could be considered a constant.
Then why would they include x3 as a state variable, and why would they discuss it's derivative?
"The thrust generated is assumed to be proportional to dm/dt" If m was constant that would be zero.

Also, if a system is non-linear, you can't just make it linear. That's what non-linear means. It is common practice to approximate non-linear systems with linear models though. Is that what you're asking about?

PainterGuy
DaveE said:
You said you thought that x3 = m could be considered a constant.
Then why would they include x3 as a state variable, and why would they discuss it's derivative?
"The thrust generated is assumed to be proportional to dm/dt" If m was constant that would be zero.

Thank you!

I don't remember much of the control theory but let's try it. I can still figure out the basics.

IMHO, I don't think that the problem is worded properly.

A rocket is propelled forward by a thrust force equal in magnitude, but opposite in direction, to the time-rate of momentum change of the exhaust gas accelerated from the combustion chamber through the rocket engine nozzle. This is the exhaust velocity with respect to the rocket, times the time-rate at which the mass is expelled, or in mathematical terms: ${T}={v}\frac{dm}{dt}$

where T is the thrust generated (force), $\frac{dm}{dt}[\itex] is the rate of change of mass with respect to time (mass flow rate of exhaust), and v is the velocity of the exhaust gases measured relative to the rocket. Source: https://en.wikipedia.org/wiki/Thrust#Examples I don't think [itex]\overset{\cdot }{m}$ and $m$ are the same things as implied in the question statement; "m" is mass and m-dot is the amount or mass of exhaust gases released per unit time. Please let me know if I've it right. I understand, as the fuel is burned, the rocket is supposed to 'lose' overall mass.

DaveE said:
Also, if a system is non-linear, you can't just make it linear. That's what non-linear means. It is common practice to approximate non-linear systems with linear models though. Is that what you're asking about?

Yes, that's what I'm asking about. I was more interested in the differential equation but looking at the differential equation from practical example would be more productive and helpful. So, how do we proceed? Thanks.

PainterGuy said:
I don't think that the problem is worded properly.
Most HW problems are a bit confusing. I think it takes a lot of knowledge and effort to write these well. Unfortunately, many profs don't put in that effort.

PainterGuy said:
I don't think m⋅ and m are the same things as implied in the question statement; "m" is mass and m-dot is the amount or mass of exhaust gases released per unit time.
Yes, they aren't the same. Like distance and speed, for example. But no, I don't think they said they were the same. Maybe you should reread that sentence. It has the form "x" is distance and dx/dt is the amount of distance covered per unit time. Personally, I don't think those are confusing or equivalent.
PainterGuy said:
Yes, that's what I'm asking about. I was more interested in the differential equation but looking at the differential equation from practical example would be more productive and helpful. So, how do we proceed?
That state space equation is a differential equation, or at least a set of equations. You could multiply it all out and reduce the number of equations with substitutions. However, everyone that does this stuff thinks it's easier in the matrix form, a series of first order equations as opposed to one higher order DE.

This subject is much too complicated to cover in a forum like this. Non-linear dynamics is usually a graduate level subject. The problem is that there are lots of types of non-linear systems. You can linearize them with a bunch of first derivatives, or perturbation theory, but then the question immediately arises, "is your model any good?" You may find that you can't get one good model that works over the range of interest.

I did study this stuff decades ago, but don't remember enough to teach it. You can look on YouTube for some Cornell lectures from Steven Strogatz, which are excellent, but they aren't a cookbook approach. It looks like there are a bunch of useful links from a google search for "linearization in modern control theory". Also Wikipedia has some good stuff.

PainterGuy
I remember this problem given in a textbook, but I cannot remember the textbook. Where was this problem taken from, please.

mpresic3 said:
I remember this problem given in a textbook, but I cannot remember the textbook. Where was this problem taken from, please.

I was helping someone with the differential equation part. After your post I asked about the book. It's linear system theory by chen.

Thanks

DaveE said:
Yes, they aren't the same. Like distance and speed, for example. But no, I don't think they said they were the same. Maybe you should reread that sentence. It has the form "x" is distance and dx/dt is the amount of distance covered per unit time. Personally, I don't think those are confusing or equivalent.
They said, "The thrust generated is assumed to be proportional to $\overset{\cdot }{m}$, where m is the mass of module."

To me, the statement is saying that that thrust is proportional to the derivative of mass 'm' where 'm' is the mass of module. But, IMHO, the derivative is being taken of mass of gases being emitted at any given time and not of the total mass of module. You see my confusion.

On the other hand, the gas tank is part the module so as the mass of gas tank decreases so does the total mass of module. Perhaps, they are looking at it, in ideal terms, where mass of module is considered separate from the mass of gas tank, and as the gas tank loses mass the module mass still remains constant.

I'd really appreciate if you can comment on it. Thanks in advance!

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PainterGuy said:
the derivative is being taken of mass of gases being emitted at any given time and not of the total mass of module.
I'm not sure I understand what you meant, but this sounds a bit like a second derivative.

PainterGuy said:
the gas tank is part the module so as the mass of gas tank decreases so does the total mass of module. Perhaps, they are looking at it, in ideal terms, where mass of module is considered separate from the mass of gas tank, and as the gas tank loses mass the module mass still remains constant.
Suppose you rocket has a mass described as m ≡ Mrocket + mfuel ≡ Mr + mf. In this model Mrocket is a constant value, and mfuel is being burned. Then I think you can see that their model (the rate that fuel is being used is proportional to thrust) makes sense. So Thrust ∝ dmf/dt.

So now look at dm/dt = d(Mr + mf)/dt = dMr/dt + dmf/dt = dmf/dt, because the derivative of a constant is zero. Thrust isn't about how much mass there is, it's about how much it is changing.

PainterGuy

## 1. Why do some differential equations have non-linear terms?

Non-linear terms in a differential equation arise when the dependent variable and its derivatives appear in a non-linear way, such as in an exponential or trigonometric function. This can happen when the system being modeled is complex and cannot be accurately described by a linear equation.

## 2. How do non-linear terms affect the behavior of a differential equation?

Non-linear terms can make a differential equation more difficult to solve and can lead to more complex behavior, such as multiple solutions or chaotic behavior. They can also cause solutions to be sensitive to initial conditions, making it difficult to predict the long-term behavior of the system.

## 3. Can a non-linear differential equation be solved analytically?

In most cases, non-linear differential equations cannot be solved analytically and require numerical methods to approximate a solution. However, there are some special cases where non-linear equations can be solved analytically, such as the Bernoulli differential equation.

## 4. How are non-linear differential equations used in science?

Non-linear differential equations are used to model a wide range of complex systems in science, including population dynamics, chemical reactions, and fluid dynamics. They allow scientists to study and understand the behavior of these systems and make predictions about their future behavior.

## 5. Are there any advantages to using non-linear differential equations over linear ones?

Non-linear differential equations can provide a more accurate representation of complex systems compared to linear equations. They can also capture more complex behavior, such as oscillations and bifurcations, that cannot be described by linear equations. However, they can be more difficult to solve and may require numerical methods to find solutions.

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