y prime prime prime prime

This can be readily solved via a Laplace Transform given our boundary conditions at $t = 0.$ This computes to:

$L(y(t)) = [s^4Y(s) - s^3f(0) - s^2f'(0) -sf''(0) - f'''(0)] - Y(s) = 0;$

or $(s^4 - 1)Y(s) - s^2 = 0;$

or $Y(s) = \frac{s^2}{s^4 - 1} = \frac{s^2}{(s^2 + 1)(s+1)(s-1)} = \frac{1}{2(s^2 + 1)} - \frac{1}{4(s+1)} + \frac{1}{4(s-1)};$

or $y(t) = L^{-1}(Y(s)) = \frac{1}{2} \sin(t) - \frac{1}{4}e^{-t} + \frac{1}{4}e^{t}.$

At $t = \pi$ we obtain:

$y(\pi) = \frac{1}{2} \sin(\pi) - \frac{1}{4}e^{-\pi} + \frac{1}{4}e^{\pi} = \frac{1}{2} \sinh({\pi}) \approx 5.756$

and $\lceil 5.756 \rceil = \boxed{6}.$