Solve the initial value problem.
-$$\frac { \mathrm { d } ^ { 2 } \mathrm {~s} } { \mathrm { dt } ^ { 2 } } = - 128 \cos ( 8 \mathrm { t } ) , \mathrm { s } ^ { \prime } ( 0 ) = 8 , \mathrm {~s} ( 0 ) = 8$$
A) s = 16 cos (8t) + 8t - 8
B) s = 2 cos (8t) + 6
C) s = 2 cos (8t) + 8t + 6
D) s = -16 sin (8t) + 8

Question

Quizplus · Accepted Answer

Taking the antiderivative twice, we get:
$$\frac { \mathrm { d } ~\mathrm {s} } { \mathrm { dt } } = 16 \sin ( 8 \mathrm { t } ) + C_1 ,$$
$$\mathrm {s} = - 2 \cos ( 8 \mathrm {t} ) + C_1 \mathrm {t} + C_2 .$$
Using the initial conditions, we get:
$$8 = C_1 ,$$
$$8 = - 2 + C_2 ,$$
$$C_2 = 10 .$$
Therefore, the solution to the IVP is:
$$\mathrm {s} = 2 \cos ( 8 \mathrm {t} ) + 8 \mathrm {t} + 6 .$$

Solve the Initial Value Problem A) S = 16 Cos (8t) + 8t - 8