Driven Harmonic Oscillator - Mathematical Manipulation of Equations

[phyzmatix]

1. Homework Statement and the attempt at a solution

Please see attached.

I'm not so sure if my problem lies with the physics or the mathematics. I have the distinct feeling that it's the latter and that I'm missing something elementary, but truly have no idea how to proceed.

Any advice will be appreciated.
Thanks!
phyz

 

[chrisk]

Express the forcing function in terms of

e^{i\omega\mbox{t}

then separate the real and imaginary parts of the solution for x. The real part is the desired solution.

 

[phyzmatix]

I'm afraid I don't quite know what you mean...How do I write

F(t)=F_0 \sin{\omega t}

in terms of

e^{i\omega t}

?

 

[chrisk]

Use the Euler Equation

e^{i\theta}=\cos{\theta}+i\sin{\theta}

and

e^{-i\theta}=\cos{\theta}-i\sin{\theta}

If you are not familiar with this, try using the identity for sin(A+B) and cos(A+B).

 

[phyzmatix]

I really appreciate the help, but please bear with me as I try to wrap my head around this. I do know the Euler equation, but my understanding is that only the real part relates to SHM and since x as well as F(t) are given as functions of sine (not cosine), I don't know how to "bridge the gap" so to speak. I've tried using the identities for sine and cosine as you mention, but end up with massively intimidating equations involving \sin\phi and \cos \phi which doesn't really help as I don't know how to get rid of either...

 

[chrisk]

After using the sin(A+B) and cos(A+B) identities, equate the coeffecients:

The sin(omega*t) coefficients on the right side of the equation are equal to F₀ and the cos(omega*t) coefficients are equal to zero.

 

[phyzmatix]

Finally the light! Thank you! I'm going to play with this and hopefully I won't get stuck again