Behavior of DE as t approaches infinity

[1MileCrash]

Homework Statement

y' = -2 + t - y

Draw a slope field and determine behavior as t -> infinity

Homework Equations

The Attempt at a Solution

This is the first DE in my book that includes t in the differential equation. The slope field looks pretty wacky. For the others I was able to see the behavior easily through the slope field because it didn't change along the t axis. Now, just looking at the slope field won't do.

How does one determine the behavior as t approaches infinity here? (without solving the equation!)

Thanks!

 

[QED Andrew]

Is $y$ constant as $t$ increases? If so, then $y'$ increases as $t$ increases; furthermore, $y'$ increases without bound as $t$ increases without bound.

 

[HallsofIvy]

Well, no, y is NOT constant as t changes! The whole point of a differential equation is that y is a function of t. And how could y' increase (or be anything but 0) if y is constant?

1MileCrash, what does the slope field look like for large t? If it looks, say, like a straight line, try y= Ax+ B and put into the equation to find A and B.

 

[tt2348]

Well think in terms Of equilibrium points. We know that y is max/min/or saddle at y'=0, so your max or min would fall on y=t-2, which would imply y goes to infinity as t does

 

[1MileCrash]

Well, no, y is NOT constant as t changes! The whole point of a differential equation is that y is a function of t. And how could y' increase (or be anything but 0) if y is constant?

1MileCrash, what does the slope field look like for large t? If it looks, say, like a straight line, try y= Ax+ B and put into the equation to find A and B.

How should I draw a slope field for large t? What should I do with the y's? Make them large too? Leave them the same? Couldn't the slope field be any number of things for large t depending on what y's I consider?

Thanks again.

 

[QED Andrew]

Well, no, y is NOT constant as t changes! The whole point of a differential equation is that y is a function of t. And how could y' increase (or be anything but 0) if y is constant?

1MileCrash, what does the slope field look like for large t? If it looks, say, like a straight line, try y= Ax+ B and put into the equation to find A and B.

Perhaps I shouldn't have phrased whether $y$ varies or not as a question. Of course $y$ varies; however, what can one say about the general behavior of $y'$ as $t$ increases without bound without restricting $y$ to a constant value?

You ask how $y'$ would increase if $y$ were constant. $y' = -2 + t - y$ is given. Let $y = y_0$ for some constant $y_0$. Then, $y' = -2 + t - y_0$, from which it is clear $y'$ is an increasing function of $t$.

Alternatively, consider the slope field for the given differential equation. Restricting $y$ to a constant value $y_0$ is equivalent to restricting the slope field to only those points on the line $y = y_0$. The restricted slope field shows $y'$ is an increasing function of $t$.

How should I draw a slope field for large t? What should I do with the y's? Make them large too? Leave them the same? Couldn't the slope field be any number of things for large t depending on what y's I consider?

Thanks again.

In answer to your last question, yes. The slope at any given point on the slope field depends on both $t$ and $y$. The slope field calculator hosted by Rutgers at http://www.math.rutgers.edu/~sontag/JODE/JOdeApplet.html might be useful to you.