The displacement x(t) of an undamped, forced mechanical oscillator is given by x ( t ) = 8 cos ( 4 t ) − 4 sin ( 4 t ) + 8 t sin ( 4 t )

Question

Find dx / dt and d^2x / dt^2

Answer 1

dx/dt = d/dt(8cos(4t) - 4sin(4t) + 8tsin(4t)) = -32sin(4t) - 16cos(4t) + 8(sin(4t) + 4tcos(4t))

dx/dt = -24sin(4t) - 16cos(4t) + 32tcos(4t)

d2x/dt2 = d/dt(-16cos(4t) - 24sin(4t) + 32tcos(4t)) = 64sin(4t) - 96cos(4t) + 32(cos(4t) - 4tsin(4t))

d2x/dt2 = -64cos(4t) + 64sin(t) - 128tsin(4t)