I'm new to differential equations, so any help will be grateful.
I've been looking at this problem:
Examine the slope field of the following differential equation. Based on the direction field, determine the behavior of $y$ as $t→∞$.
$$y' = -2 -y + t$$
After plotting the slope field, I could not seem to deduce anything, so curiously, I looked at the solution and this is what it stated:
y is asymptotic to $t − 3$ as $t→∞$.
Could someone explain to me what they mean by "asymptotic to …" My textbook didn't give me an example regarding this type of question so I do not understand what it means.
NOTE
I've noticed that the solution to this differential equation is:
$$y = c_1e^{-t} + t -3$$
$t-3$ is apparent in the solution.
Could this be in connection with the solution that they provided?
Again help will be appreciated. Thanks in advance.
Best Answer
$y = c_1e^{-t} + t -3$
Notice the negative exponent makes the first term vanish as $t\to \infty$ and the solution almost looks same as $y=(\approx 0) + t-3$
slope field gives the same picture :