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: For the linear first-order ODE y' + 2ry = t², use integrating factor e^(2rt) to obtain d/dt(ye^(2rt)) = t²e^(2rt). As t→∞, the exponential term dominates, so asymptotically y ~ t/(2r) by comparing the growth rates of y and t.
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