Construct ODE that approaches an asymptote

• usn7564
In summary, the equation describes what y(t) should be, but x(t) needs to be connected to it in order to get the solution.
usn7564

Homework Statement

Construct a first order linear differential equation whose solutions have the required behavior as t approaches infinity. Then solve your equation and confirm that the solutions do indeed have the specified property.

All solutions are asymptotic to the line y = 2 - t as t approaches infinity.

I don't even know where to begin, could someone give me a kick in the right direction? I tried just writing a linear differential equations with two unknown functions f and g (y' + fy = g) and getting the general solution but obviously that involves a completely unknown integral and I doubt it would lead anywhere.

usn7564 said:

Homework Statement

Construct a first order linear differential equation whose solutions have the required behavior as t approaches infinity. Then solve your equation and confirm that the solutions do indeed have the specified property.

All solutions are asymptotic to the line y = 2 - t as t approaches infinity.

Let $x(t) = y(t) - (2-t)$.

Can you think of a linear first-order ODE with constant coefficients for which all solutions tend to zero as $t \to \infty$?

Now let $x$ satisfy that ODE.

Thanks, could use a few more pointers though. So I put x(t) = y(t) - (2-t) where x approaches zero as t approaches infinity which makes sense as the equation describes what y(t) I want.
Can't see how I relate it to the ODE though. Say I find one where all solutions tend to zero (y' + y = 0) for example, how do I connect it to x(t)?

usn7564 said:
Thanks, could use a few more pointers though. So I put x(t) = y(t) - (2-t) where x approaches zero as t approaches infinity which makes sense as the equation describes what y(t) I want.
Can't see how I relate it to the ODE though. Say I find one where all solutions tend to zero (y' + y = 0) for example, how do I connect it to x(t)?

Take $x' + x = 0$ and substitute $x = y - (2-t)$.

pasmith said:
Take $x' + x = 0$ and substitute $x = y - (2-t)$.
Understand now, thank you. No idea why it was so tricky to wrap my head around it now, but there you go.

1) What is an asymptote?

An asymptote is a line that a curve approaches but never touches. It can be horizontal, vertical, or oblique.

2) How do you construct an ODE that approaches an asymptote?

To construct an ODE (ordinary differential equation) that approaches an asymptote, you need to manipulate the equation to have a form where the dependent variable approaches infinity or zero as the independent variable approaches a certain value. This value represents the asymptote.

3) What are the key components of an ODE that approaches an asymptote?

The key components of an ODE that approaches an asymptote are the dependent variable, the independent variable, and the asymptote value. The equation should also include terms that approach infinity or zero as the independent variable approaches the asymptote value.

4) Can an ODE have multiple asymptotes?

Yes, an ODE can have multiple asymptotes. This can happen when there are multiple values that the dependent variable approaches as the independent variable approaches different values.

5) How can an ODE that approaches an asymptote be used in real-world applications?

ODEs that approach asymptotes can be used to model situations in which a certain value is never reached, but rather the system approaches it asymptotically. Some examples include population growth and radioactive decay. These types of equations can also be used to predict future behavior of a system based on its current behavior.

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