(c) y(0)=2Yo, y '(0) = 0; (b) y(0) = 0, y'(0) = Vo;

(d) y(0) = 0, y'(0) = 20;

Solved Step by step with explanation: find y(t) and specify their amplitude and natural frequency.

1. Find the amplitude:

The amplitude (A) is equal to the initial displacement, so A = Yo.

y''(t) + ky(t) = 0.

The characteristic equation for this differential equation is:

r = ±i*sqrt(k).

The general solution for SHM is given by:

y'(0) = -Aωsin(0) + Bωcos(0) = Bω = 0.

From the second equation, we have B = 0 (since ω is nonzero).

1. Find the amplitude:

Since the initial displacement is zero, the amplitude (A) is also zero.

y''(t) + ky(t) = 0.

The characteristic equation for this differential equation is:

r = ±i*sqrt(k).

The general solution for SHM is given by:

y'(0) = -Aωsin(0) + Bωcos(0) = Bω = Vo.

From the second equation, we have B = Vo / ω.

1. Find the amplitude:

The amplitude (A) is equal to the initial displacement divided by 2, so A = 2Yo / 2 = Yo.

Assuming m = 1 (to simplify the equation without loss of generality), we have:

y''(t) + ky(t) = 0.

r = ±sqrt(-k),

r = ±i*sqrt(k).

y(0) = Acos(0) + Bsin(0) = A = 2Yo,

y'(0) = -Aωsin(0) + Bωcos(0) = Bω = 0.

For this case, the initial displacement is zero, and the initial velocity is 20.

1. Find the amplitude:

Assuming m = 1 (to simplify the equation without loss of generality), we have:

y''(t) + ky(t) = 0.

r = ±sqrt(-k),

r = ±i*sqrt(k).

y(0) = Acos(0) + Bsin(0) = A = 0,

y'(0) = -Aωsin (0) + Bωcos(0) = Bω = 20.