Disagreement with my teacher about linear DE's

In summary, the conversation revolves around the concept of linearity in differential equations. The speaker proposes a new way of understanding linearity by considering the independent variable as a constant and the variables and derivatives as standard variables. They also discuss the difference between linear and nonlinear equations and how this idea fits into the concept of linearity. The validity of this idea is questioned, but the speaker explains that it is just an interesting thought that arose during their discussion on the origins of the term "linearity."
  • #1
1MileCrash
1,342
41
I think I may have just phrased what I meant poorly but..

If I take any linear DE, consider the independent variable (t) to be some constant, and consider y and all its derivatives to be just standard variables, and I graph this function, it always is a linear object (line, plane, etc) with dimension depending on how many y/derivatives of y are in the differential equation.

I don't see how this could be false. This is a direct consequence of y and all derivatives having a linear relation, which is the definition of a linear DE.
 
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  • #2
t can't be considered a constant, considering it to be a constant would cause all meaning to be lost from the derivatives.

Though, if I'm getting you correctly, you're replacing t with c (where c is an arbitrary constant) and f, f', f'', etc. with with independent variables and seeing if the resulting equation is linear wrt f, f', f'', and the other derivatives? I'm pretty sure it will be, but I don't really see why there's any reason to do this, especially since you just get meaningless jargon (again, due to the fact that the derivatives now mean nothing.) No offence, though, interesting idea!
 
  • #3
Whovian said:
t can't be considered a constant, considering it to be a constant would cause all meaning to be lost from the derivatives.

Though, if I'm getting you correctly, you're replacing t with c (where c is an arbitrary constant) and f, f', f'', etc. with with independent variables and seeing if the resulting equation is linear wrt f, f', f'', and the other derivatives? I'm pretty sure it will be, but I don't really see why there's any reason to do this, especially since you just get meaningless jargon (again, due to the fact that the derivatives now mean nothing.) No offence, though, interesting idea!

Yes, this is what I mean.

I know that t can't "actually" be considered constant, but for the sake of attributing meaning to why the term "linear" is used I thought of it as being "ignored" more so than constant.

He stated that in math we use words that don't always fit (linear differential equation) but with this process it fits perfectly, although I'm not very good at expressing my ideas.

Like..
cos(yy') = 2t is nonlinear, and the graph of

Y = arccos(C)/X = K/X

Is not a line , where X is Y', C is a constant for 2t, and K is arccos(C) which is some other constant.

Meanwhile

2y' + ty = 0

Is linear and

Y = -2X/C

Is a line with the same changes as before.
 
  • #4
2y' + ty = 0
is linear in the in the sense that if
y1 and y2 are solutions
c1y1+c2y2 is a solutions
 
  • #5
lurflurf said:
2y' + ty = 0
is linear in the in the sense that if
y1 and y2 are solutions
c1y1+c2y2 is a solutions

This is true but I don't see what this has to do with the validity of what I am describing.
 
  • #6
I do not know what you are describing. Linear DE's are linear, that is why they are so named. Perhaps you are talking about the fact that what some call affine others call linear.

If y1 and y2 are solutions
then c1y1+c2y2 is a solutions
is what linear means

(if Ly=f instead of Ly=0
If y1 and y2 are solutions
then (c1y1+c2y2)/(c1+c2) is a solutions
is what linear means
 
Last edited:
  • #7
lurflurf said:
I do not know what you are describing. Linear DE's are linear, that is why they are so named. Perhaps you are talking about the fact that what some call affine others call linear.

What I am describing is just an "idea" that arose when we were discussing the origins of the term linearity.

I understand that linear DEs are linear regardless of this idea (for the exact reasons you said) I am just providing an origin for what led me to think about it. I am not asking what makes a linear DE linear, I'm asking if the idea I've outlined in my first post is a true one.
 

1. What are linear differential equations (DE's)?

Linear differential equations are mathematical expressions that describe the relationship between a function and its derivatives. They are called linear because they only involve the function and its derivatives with respect to the independent variable, and they do not involve any higher powers or products of the function or its derivatives.

2. Why would I disagree with my teacher about linear DE's?

It is possible to have different interpretations or approaches when solving linear DE's, and it is not uncommon for students to have different methods or solutions compared to their teacher. Additionally, teachers may have different preferences or techniques when teaching linear DE's, which can lead to disagreement.

3. How do I know if I am solving linear DE's correctly?

There are several ways to check if you are solving linear DE's correctly. One way is to check if your solution satisfies the original differential equation. Another way is to compare your solution with your teacher's or with a known solution. Additionally, checking for consistency and simplifying your solution can also help ensure accuracy.

4. Can linear DE's have more than one solution?

Yes, linear DE's can have multiple solutions. In fact, there are typically infinitely many solutions for a given linear DE. This is because when we solve a linear DE, we often end up with a general solution that includes a constant or arbitrary value. Therefore, there are many possible combinations of constants that can result in different solutions.

5. How can I improve my understanding of linear DE's if I am having a disagreement with my teacher?

If you are struggling to understand linear DE's or are having a disagreement with your teacher, it may be helpful to seek additional resources such as textbooks, online tutorials, or tutoring services. You can also try discussing your approach with your teacher and asking for clarification on specific concepts or steps. Practice and persistence can also help improve your understanding of linear DE's.

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