30) Find the value of y_{0} 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?