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