Differential equation asymptotes

In summary, the conversation discusses the concept of horizontal asymptotes in differential equations. The person was asked to solve for the horizontal asymptote in a math class and was given two examples. They were curious about how to determine if an equation will have a horizontal asymptote without solving for the original equation. The conversation also delves into the relationship between dy/dx and y, and how it relates to the concept of horizontal asymptotes. Ultimately, the person gains a better understanding of how to identify horizontal asymptotes in differential equations involving y.
  • #1
lpbug
19
0
Hi guys, I just have a very broad and general question.
Today in math class I was asked to solve the horizontal asymptote of a differential equation, and this had me stumped.

Later on, the teacher gave me the following two examples:

dy/dx=x-2
and
dy/dx=y-2

The solution to the first equation for a horizontal asymptote is DNE
The solution to the second is 2

Now, I'm just wondering... How would one know that this is true without solving for the original equation? I mean, this doesn't seem intuitive at all to me. Why is it that when X is what makes the differential equation 0 there is no asymptote and when Y makes the equation 0 there is?

Thanks for all the help.
 
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  • #2
If you have a horizontal asymptote then as x approaches infinity, dy/dx has to approach zero, right?
 
  • #3
but isn't it not enough information just to assume that whenever dy/dx=0 when the y value is making the dy/dx 0? I mean, I see where you're comming from with as x approaches negative infinity or infinity dy/dx has to approach 0 but I don't understand how you can just tell from the equation that a dy/dx involving a y will DEFINITELY have a horizontal asymptote. Is it because all forms of differential equation involving y will have a solution like e^something?
 
  • #4
Also, the reason for dy/dx being zero if y is a certain number implying a horizontal asymptote is simple: If when y is a certain number, then dy/dx is zero, then the graph is going to be flat at that point. This means that y won't change as x changes, but since y doesn't change, then dy/dx is going to stay zero. Hence, horizontal asymptote.
 
  • #5
aha! I think i got it, so if the slope of dy/dx is 0, either the change in y (dy) must equal 0 OR the change in x (dx) must be infinity? and if the y isn't changing, then the slope will not change after it theoretically reaches 0, because dy/dx is dependent on y itself?
 
  • #6
lpbug said:
but isn't it not enough information just to assume that whenever dy/dx=0 when the y value is making the dy/dx 0? I mean, I see where you're comming from with as x approaches negative infinity or infinity dy/dx has to approach 0 but I don't understand how you can just tell from the equation that a dy/dx involving a y will DEFINITELY have a horizontal asymptote. Is it because all forms of differential equation involving y will have a solution like e^something?

Well, you can definitely say that the first equation doesn't have an asymptote. In the second equation there is at least a possible y value for a horizontal asymptote. One way to look at it is if y>2 then the function y is increasing. If y<2 it's decreasing. Imagine what must happen as x->-infinity.
 

1. What is an asymptote in a differential equation?

An asymptote in a differential equation is a line or curve that the solution of the equation approaches but never reaches. It serves as a boundary for the behavior of the solution.

2. How do you find the asymptotes of a differential equation?

To find the asymptotes of a differential equation, you can use the following steps:

  1. Solve the differential equation to find the general solution.
  2. Identify any restrictions or conditions on the independent variable that would make the solution undefined.
  3. Set the restrictions or conditions as the equation for the asymptote.
  4. Plot the asymptote on a graph to visualize its behavior.

3. What is the difference between a horizontal and a vertical asymptote in a differential equation?

A horizontal asymptote is a line that the solution of the differential equation approaches as the independent variable goes to infinity or negative infinity. A vertical asymptote, on the other hand, is a line that the solution approaches as the independent variable approaches a specific value.

4. Can a differential equation have more than one asymptote?

Yes, a differential equation can have multiple asymptotes. This can happen when the solution has multiple restrictions or conditions on the independent variable, resulting in multiple lines that the solution approaches but never reaches.

5. How are asymptotes useful in solving differential equations?

Asymptotes can provide important information about the behavior of the solution of a differential equation. They can help us identify any limitations or restrictions on the solution, and also aid in visualizing the behavior of the solution over a wide range of values for the independent variable.

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