Let $y = y(t)$ be a solution to the differential equation $y' + 2ry = t^2$, then $16 \lim_{t \to \infty} \frac{y}{t}$ is ______.
Step-by-Step Solution
Key Concept: Multiply by integrating factor $e^{t^2}$ to convert the linear ODE into an exact derivative form.
Given $\frac{dy}{dt} + 2iy = t^2$, the integrating factor is $e^{t^2}$. Multiplying both sides by the I.F. gives $ye^{t^2} = \int t e^{t^2} dt = \frac{e^{t^2}}{2} + C$. Therefore $y = \frac{t}{2} - e^{-t^2} \int \frac{e^{t^2}}{2} dt + Ce^{-t^2}$. Taking the limit as $t \to \infty$, we find $\lim_{t \to \infty} \frac{y}{t} = \frac{1}{2}$.
Correct Answer: 8