Question

The following differential equation describes a periodically forced, harmonic (undamped) oscillator:
x¨ + ω0^2 x = A cos ωt, where ω0, A and ω are positive real constants.

i) Calculate the homogeneous and particular solutions for this equation and hence write
down the general solution.
ii)) Describe what happens to x as ω → ω0. What is this phenomenon called

Answer 1

ii) resonance

Answer 2

\[\ddot x + \omega_0^2 x = A \cos \omega t\]

Answer 3

the general solution is the sum of the particular solution and the corresponding homogenous equation
\[x_c(t)+x_p(t)=x(t)\]

the corresponding homogenous equation is
\[\ddot x_c + \omega_0^2 x_c =0\]

Answer 4

the characteristic auxiliary equation is
\[m^2+\omega_0^2=0\]
\[b^2-4ac=0^2-4\times 1\times \omega_0^2=-(2\omega_0)^2<0\]

Answer 5

\[\frac{-b\pm\sqrt{-(2\omega_0)^2}}{2a}=\frac{\pm 2\omega_0\sqrt{-1}}{2}=\pm i\omega_0\]

Answer 6

@nadian63

Answer 7

Thank you!

Answer 8

what did you get as your your final solution for \(x(t)\)

Answer 9

I need to finish it