Question

write solution of x''+9x = sin(2t) x(0)=0 and x'(0)=1 as the sum of two oscillations

Answer 1

The complementary solution is
x = Asin(3t) + B cos(3t) or C*sin(3t + φ)

The particular integral would be of the form
x = a*sin(2t)
x" = -4a*sin(2t)
x" + 9x = 5a*sin(2t)
a = 1/5

Final solution:
x = C*sin(3t + φ) + (1/5) * sin(2t)

First oscillation is C*sin(3t + φ)
second is (1/5) * sin(2t)