# Solving this second order differential equation (Damping mechanism)

I'm trying to self-teach myself differential equations, but I'm having trouble with second order equations! How can I solve this one?

$\frac{3}{32}y''+12y=0$, $y(0)=\frac{-1}{12}$, $y'(0)=2$

Thanks! I'd appreciate if you show step by step so that I can understand it for the rest of the problems.

Consider this to be a damping mechanism now... What would be the bounds on the damping force coefficient k so that oscillatory motion would remain intact?

• Nevermind! I figured it out. – Eric Johnson Dec 9 '14 at 20:56
• This question appears to be off-topic because the original poster no longer needs the answer and there are already similar questions. – Joonas Ilmavirta Dec 9 '14 at 21:01
• @JoonasIlmavirta Wait, nevermind. I can't figure out the IVP part – Eric Johnson Dec 9 '14 at 21:02
• This question might also provide an answer. – Milo Brandt Dec 9 '14 at 23:07
• I originally voted to close this because of @JoonasIlmavirta's comment, but the OP has since signaled he still wanted help. So, I'm voting to reopen. – apnorton Jan 21 '15 at 18:31

I will map out the steps, but please fill in the details.

Using Undetermined Coefficients, we have:

$$\dfrac {3}{32} m^2 + 12 = 0 \implies m_{1,2} = \pm ~8i \sqrt{2}$$

This means our solution is:

$$y(t) = c_1 \cos (8 \sqrt{2} t) + c_2 \sin(8 \sqrt{2} t)$$

Now, find $y'(t)$ and substitute $y'(0)$ and $y(0)$ to solve for $c_1$ and $c_2$.

You should get (hover over area to see spoiler):

$$y(t) = -\dfrac{1}{12} \cos(8 \sqrt{2} t) + \dfrac{\sqrt{2}}{8} \sin(8 \sqrt{2} t)$$

• how did you do that "hover over area"? What's the format?? – Eric Johnson Dec 9 '14 at 21:25
• What about bounds on the damping force coefficient if oscillatory motion is to be conserved – Eric Johnson Dec 9 '14 at 21:34
• A mass weighing 3 lbs stretches a spring 3 inch. skipping the initial conditions part.... Find the position of mass at any time. IF a damping force was introduced, what bounds on the damping constant would be needed so that its oscillatory motion conserved. – Eric Johnson Dec 9 '14 at 21:42
• Okay, also, how do i get te answer in form y(t) = Rcos(wt-phi) ? – Eric Johnson Dec 9 '14 at 21:47
• huh i dont know how i could do that – Eric Johnson Dec 11 '14 at 3:14