Question Regarding Homework Assignment 2.1 #30

[Zeya Chen]

30) Find the value of y₀ for which the solution of the initial value problem
$$
y′−y=1+3\sin(t)\qquad y(0)=y_0
$$
remains finite as $t\to \infty$.

It's clearly to see that the integrating factor is $e^{-t}$.        OK. V.I.

Then $y(t) = e^{t}\int  e^{-t } \bigl(1 + 3\sin(t)\bigr)\,dt + ce^t$,                      I fixed it
which can be easily solved as implies
$$
y(t) = - 1 - \frac{3}{2}\bigl(\sin(t) + \cos(t)\bigr) + ce^t
$$
But how can we interpret the term "remains finite as $t\to \infty$" into algebraic language  for solving this initial value problem?

[Victor Ivrii]

What should be $c$ in order to $y(t)$ be bounded as $t\to +\infty$? In this case $y(0)=?$

[Zeya Chen]

Thank you for your reply Prof Ivrii.

c have to be zero since e^t is an positive increasing function of t.

[Victor Ivrii]

Please look how I modified your post. This is how mathematics should be typed. Also it is not important that $e^t$ is increasing but that it is unbounded as $t\to \infty$.