General Solution as Trigonometric Functions and in Complex form

[JoshWalker]

While I would like assistance with the first part of the question my main issue is expressing it in trigonometric and complex forms. If possible any clarification on the term general solution and what it refers to would also be appreciated.

A harmonic oscillator is described by the equation
d^2x/dt^2 +9x = 0
with time measured in seconds.

a) What is the angular frequency, in Hertz, of the oscillator?

b) Find the general solution of d^2x/dt^2 +9x = 0, first expressed with trigonometric functions, then in complex form, as in equation.

 

[MarkFL]

The general solution, in this case as we have a second order ODE, will be the two-parameter family of functions satisfying the given ODE. The characteristic/auxiliary equation is:

[MATH]r^2+9=0[/MATH]
Hence the characteristic roots are:

[MATH]r=\pm3i[/MATH]
From this, what would then be the general solution in trigonometric form?

 

[MarkFL]

To follow up:

Based on the characteristic roots, the general solution may be given as:

[MATH]x(t)=c_1\cos(3t)+c_2\sin(3t)[/MATH]
And so the angular frequency is:

[MATH]f=\frac{3}{2\pi}\text{ Hz}[/MATH]
The solution in complex exponential form is:

[MATH]x(t)=c_1e^{3it}+c_2e^{-3it}[/MATH]