Mechanical ad Electrical vibration EASY question ?

[dwilmer]

Mechanical ad Electrical vibration EASY question please help?

In Boyce and Diprima textbook it says that:

mu'' + ku = 0 .

Then it says the general solution of this is:

u = Acos (w^2)t + B sin (w^2)t , where w^2 = k/m


It provides no explanation how it arrives at this. Where does the cos and sin come from?

PS: it is supposed to be the greek letter w, whatever that is called and it also has a sub-0 on it, but i didnt include it because it looks confusing without the right fonts.

please help, thanks

 

[tiny-tim]

Welcome to PF!

Hi dwilmer! Welcome to PF!

(have a little omega: ω and try using the X² tag just above the Reply box )

mu'' + ku = 0 .

Then it says the general solution of this is:

u = Acos (w^2)t + B sin (w^2)t , where w^2 = k/m


It provides no explanation how it arrives at this. Where does the cos and sin come from?

Because [cos(ωt)]'' is obviously -ω²cos(ωt), and [sin(ωt)]'' is obviously -ω²sin(ωt)

 

[n1person]

Perhaps a more helpful explanation can be found here:

http://www.stewartcalculus.com/data...ntexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf

page 4 in particular offers a little bit of an explanation, and if you are still confused I would advise reading the entire thing, it isn't too long and is relatively well done.