Using the mass-spring analogy, predict the behavior as $t \to \infty$ of the solution to the given initial value problem. Then confirm your prediction by actually solving the problem.
$\begin{array}{l} (a) . y'' + 16 y = 0; y (0) = 2, y' (0) = 0 \\ (b) . y'' + 100 y' + y = 0; y (0) = 1, y' (0) = 0 \\ (c) . y'' - 6 y' + 8 y = 0; y (0) = 1, y' (0) = 0 \\ (d) . y'' + 2 y' - 3 y = 0; y (0) = - 2, y' (0) = 0 \\ (e) . y'' - y' - 6 y = 0; y (0) = 1, y' (0) = 1 \end{array}$

Question

Accepted Answer

The solution is y=2cos4t.
The general solution is $y (t) = \frac{- 50 + \sqrt{2499}}{2 \sqrt{2499}} e^{(- 50 + \sqrt{2499}) t} + \frac{50 + \sqrt{2499}}{2 \sqrt{2499}} e^{(- 50 - \sqrt{2499}) t}$
The general solution is $y (t) = 2 e^{2 t} - e^{4 t}$ .
The general solution is $y (t) = - \frac{1}{2} e^{- 3 t} - \frac{3}{2} e^{t}$ .
The general solution is $y (t) = - \frac{2}{5} e^{- 3 t} + \frac{3}{5} e^{t}$ .

Short Answer