# Construct ODE that approaches an asymptote

1. Sep 13, 2013

### usn7564

1. The problem statement, all variables and given/known data
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.

2. Sep 13, 2013

### pasmith

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.

3. Sep 15, 2013

### usn7564

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)?

4. Sep 15, 2013

### pasmith

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

5. Sep 15, 2013

### usn7564

Understand now, thank you. No idea why it was so tricky to wrap my head around it now, but there you go.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted