Differential Equations behavior for large t?

In summary, dy/dt = 2 - 2ty = 0 when solved for t = τ/ε. Large t corresponds to the limit ε→0, so ignoring the second order in ε you would get the behavior y = \frac{\epsilon}{\tau} = \frac{1}{t} for large t. This is approximate, but the 1/t approximation is even better. Setting dy/dt = 0 allows for the equilibrium solution as a constant, even when solving for non-autonomous equations.
  • #1
Gridvvk
56
1
dy/dt = 2 - 2ty
y(0) = 1

I am not asked to solve this (I know it's not easy to solve), but what I am asked is,

"for large values of t is the solution y(t) greater than, less than, or equal to 1/t"?

I would think less than because 1/e^(t^2) converges faster than 1/t, but at the same time I solved dy/dt = 0 and got y = 1/t, so I'm not sure what that signifies.
 
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  • #2
I am not sure how you got 1/t by solving dy/dt = 0, perhaps there is a typo?

To look at the behavior for large t, I rescaled variables to t=τ/ε. The differential equation then becomes

[itex]\epsilon^{2}\frac{dy}{d\tau} = 2\epsilon - 2\tau y[/itex].

Now large t corresponds to the limit ε→0, so ignoring the second order in ε you would get the behavior [itex]y = \frac{\epsilon}{\tau} = \frac{1}{t}[/itex] for large t.

As a check I used maple to solve the equation and then plotted it against 1/t. The convergence to 1/t seems pretty rapid actually. By the time you hit t=5 the solution differs from 1/t by only ~0.004.

I think problems like these come up when you are looking at asymptotic methods for solving differential equations. This is something I am just getting into for a summer research project so I am by no means an authority on the method. I don't really have any book recommendations for this either... but I am reading some sections of Perturbation Methods by Ali Nayfeh. You could probably find more information by searching up asymptotic methods.
 
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  • #3
dy/dt = 0 => 2 - 2ty = 0

2(1 -ty) = 0
1 - ty = 0
1 = ty
y = 1 / t

This of course assumes can't be zero.

Thanks for looking into it, just to clarify:

According to your results from maple, are you saying, it should actually be equal to 1 / t for large t?
 
  • #4
Yes, for large t it goes like 1/t. Maple was just a check though, I would base it primarily on the epsilon going to zero limit. epsilon^2 goes to zero faster than epsilon which is why I set it to zero, leaving epsilon by itself. If you set both epsilon and epsilon^2 to zero then you get an even cruder approximation for large t, mainly that y tends to 0 for large t, which is also approximately true, but the 1/t approximation is even better.

What was your reasoning for setting dy/dt = 0?
 
  • #5
Right, thanks for the help!

Well usually when we have an autonomous equation dy / dt = f(y) by setting dy/dt = 0 we get the equilibrium solution which is usually a constant, I thought the same idea could be extended to non-autonomous equations dy/dt = f(t,y), but in this case you would get the equilibrium solution as a function t; though I'm not sure if it's mathematically valid, or it may be a case of where it's bad form mathematically, but it still results in the 'correct solution'.
 

1. What is the behavior of a differential equation for large t?

The behavior of a differential equation for large t depends on its type and the values of its parameters. In general, as t approaches infinity, the solution of a differential equation will either approach a constant value, oscillate between two values, or grow without bound.

2. How can I determine the behavior of a specific differential equation for large t?

To determine the behavior of a differential equation for large t, you can analyze its stability and equilibrium points. If the equilibrium points are stable, the solution will approach a constant value. If they are unstable, the solution will oscillate or grow without bound. Additionally, you can use numerical methods such as Euler's method or Runge-Kutta method to approximate the solution for large t.

3. Can a differential equation have multiple behaviors for large t?

Yes, a differential equation can have multiple behaviors for large t. This can occur when the equation has multiple equilibrium points with different stability properties. In this case, the solution may approach different values or oscillate between different values depending on the initial conditions.

4. Is there a way to predict the behavior of a differential equation for large t without solving it?

Yes, there are certain methods for predicting the behavior of a differential equation without solving it. One approach is to analyze the direction field, which shows the behavior of the solution over time. Another method is to use phase plane analysis, which plots the solutions of a system of two differential equations in a two-dimensional space to predict their behavior for large t.

5. What are some real-world applications of studying the behavior of differential equations for large t?

Studying the behavior of differential equations for large t has many real-world applications. It is used in physics to model the motion of particles and in chemistry to understand reaction rates. In engineering, it is used to design control systems for various processes. It also has applications in economics, biology, and other fields where changes over time are important to study.

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