Consider the initial value problem

$y^{\text{'}}-\frac{9}{2}y=9t+2e^t,y(0)=y_0$

Find the value of $y_0$ that separates solutions that grow positively as $t$ from those that grow negatively. $($A computer algebra system is recommended places.$)$

$y_0=$

How does the solution that corresponds to this critical value of $y_0$ behave as $t?$

The corresponding solution will increase without bound.

The corresponding solution converges to the function $y=9t.$

The corresponding solution will decrease without bound.

The corresponding solution converges to the function $y=-9t.$

The corresponding solution converges to the function $y=0.$