Hello all!

Problem: For y'= e

^{-t} + y . Draw the direction field. And determine the behavior of y, as t approaches + infinity.

Here is the directional field:

If we notice as t-> infinity, if y>0 then y-> infinity, if y<0 y-> negative infinity.

If we notice the x-axis, y=0

Thus, all values (t,0), have positive slopes. Thus as t-> infinity y-> positive infinity if at some point y=0

However the solution says there exists an interval curve in which as t-> infinity y->0 . However, as I look at the directional fields as t-> infinity. If y>0 it goes to positive infinity eventually, if y=0 it goes to positive infinity eventually, and if y<0 it goes to negative infinity.

So what integral curve would exhibit such property that as t-> infinity y-> 0 ?

Also for this problem is there a way to analytically find which values of y(0), will give infinity, 0, or negative infinity as t-> infinity. Because graphically, we can see it is when y(0)>- 0.5 (approximate). But is there a way to do this algebraically?

Thanks.