Find the solution to the differential equation $\frac{d^{2} s}{d t^{2}}-25 s=0$ given the boundary conditions $s(0)=10$ and $s\left(\frac{s}{2}\right)=5$ .
A) $s(t)=5 \cos 25 t-10 \sin 25 t$
B) $s(t)=10 \cos 5 t+5 \sin 5 t$
C) $s(t)=10 \cos 5 t-5 \sin 5 t$
D) $s(t)=5 \cos 25 t+25 \sin 5 t$
E) $s(t)=50 \cos 5 t$

Question

Quizplus · Accepted Answer

The differential equation is a second-order linear homogeneous equation with constant coefficients. Its characteristic equation is $r^2 - 25 = 0$, which has solutions $r = \pm 5$. Thus, the general solution is $s(t) = A \cos(5t) + B \sin(5t)$. Using the boundary conditions $s(0) = 10$ and $s\left(\frac{\pi}{2}\right) = 5$, we find that $A = 10$ and $B = 5$, leading to the solution $s(t) = 10 \cos(5t) + 5 \sin(5t)$.

Find the Solution to the Differential Equation d2sdt2−25s=0\frac{d^{2} s}{d t^{2}}-25 s=0dt2d2s​−25s=0

Find the Solution to the Differential Equation $\frac{d^{2} s}{d t^{2}}-25 s=0$