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.