It is known that the displacement x(t) from the position of equilibrium of a car

Question

It is known that the displacement x(t) from the position of equilibrium of a cart that is a part of the mass-spring system is a solution of the initial value problem as follows: 36 x + x = 0, x(0) = 2, x (0) = -1. Calculate the amplitude of the oscillations of the velocity that corresponds to the displacement x(t). Rounded the value the amplitude you just found to four significant decimal digits and provide the result. Also you provide some intermediate results obtained by you while solving the problem above:

Explanation / Answer

36 x'' + x =0

characteristic equation is 36 r^2 + 1 =0

r = -1/6 i , r = 1/6 i

hence

x= a cos (t/6) + b sin (t/6)

since

x(0) - 2 , x'(0)= -1

we get a = 2 , b = -6

hence x(t) = 2 cos (t/6) - 6 sin (t/6)

velocity = x' which is

x ' = -1/3 sin (t/6) - cos(t/6)

amplitude = sqrt((-1/3)^2 +(-1)^2 ) = sqrt(10/9)

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