IVP applications of second-order ODE

1. Mar 5, 2012

zebrastripes

1. The problem statement, all variables and given/known data

Given the equation mx''+cx=cAsin(Ωt) with the initial conditions x(0)=0 and x'(0)=0.
Solve the initial value problem for the case when Ω < ω and show that |x(t)| < H provided
A < H(1-(Ω/ω)).

2. Relevant equations

3. The attempt at a solution

For my solution to the equation I got x(t)=(Aω/(ω^2-Ω^2))[ωsin(Ωt)-Ωsin(ωt)]

So, I'm hoping that this is right which I found using x(t)=x_h+x_p.
But I'm completely confused about the second part to show that |x(t)| < H provided
A < H(1-(Ω/ω)).

This is my first post here, so apologies if I'm not doing it right. :shy:
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 5, 2012

LCKurtz

Good work so far; you are almost there. Take the absolute value of both sides of your $x(t)$ equatiion and use this:$$|\omega \sin(\Omega t) - \Omega \sin(\omega t)|\le |\omega| | \sin(\Omega t)| + |\Omega| | \sin(\omega t)| \le \omega + \Omega$$and put in the overestimate given for A. See what drops out.

3. Mar 5, 2012

zebrastripes

Ahh Ok, so it's my modulus work that's really poor!

I'm thinking you're equation changes mine to: Aω/(ω^2-Ω^2)<H(ω+Ω)

So with the substitution for A I get (ω-Ω)/(ω^2-Ω^2)<(ω+Ω)

So from this, could I finish by showing this is true? Or have I gone wrong again? I'm thinking that my Aω/(ω^2-Ω^2) should have changed when I was taking the modulus?

4. Mar 5, 2012

LCKurtz

The contraction "you're" is a short form of "you are". The possessive case is "your". Also, while I'm commenting on your post, I should point out that you never told us what $\omega$ stands for. Best not to leave definitions out.

You will have less trouble if you will write out your full string of inequalities. You have$$x(t) = \frac{A\omega}{\omega^2-\Omega^2}(\omega \sin(\Omega t) - \Omega \sin(\omega t))$$Start by taking the absolute value of both sides:$$|x(t)|=\left| \frac{A\omega}{\omega^2-\Omega^2} \right| |\omega \sin(\Omega t) - \Omega \sin(\omega t)| \le ...$$Now continue the string on the right using what you know so far.