Step 1

The characteristic polynomial is 2z² + 18, and its roots are z = ±i3.

Step 2

The general solution is y = A cos (3t) + B sin (3t).

Step 3

Substitute 0 for t and 2 for y. 2 = A cos 0 + B sin 0 = A.

Compute y'(t) for the general solution.

y' = -3A sin (3t) + 3B cos (3t).

Substitute 0 for t and 15 for y'.

15 = -3A sin 0 + 3B cos 0= -3 · 0 + 3 · B, B = 5.

The particular solution, shown in Figure 14.6.3, is

y = 2 cos (3t) + 5 sin (3t).