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johnny0929
 2 years ago
Solve the initial value problem\[u''+6u'+12u=0\]with \(u(0)=1\) and \(u'(0)=0\)
johnny0929
 2 years ago
Solve the initial value problem\[u''+6u'+12u=0\]with \(u(0)=1\) and \(u'(0)=0\)

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modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3use characteristic equation a^2 + 6a+12 solve for a

abb0t
 2 years ago
Best ResponseYou've already chosen the best response.1this is a second order homogeneous differential equation. do you know the general solution? \[y_p = e^{rt}c_1+e^{rt}c_2\]

abb0t
 2 years ago
Best ResponseYou've already chosen the best response.1@modphysnoob is correct. except he used "a" so your solution would be: \[y_p = e^{at}c_1+e^{at}c_2\]

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0I actually have the answer but I am just a little confused about how to do it... It's a spring mass problem and the answer is \[u=e^{3t}cos\sqrt{3}t+\sqrt{3}e^{3t}sin\sqrt{3}t\]

abb0t
 2 years ago
Best ResponseYou've already chosen the best response.1then, plug in the initial conditions. take a few derivatives and violê

abb0t
 2 years ago
Best ResponseYou've already chosen the best response.1solve for u(0)=1 take the derivative of your particular solution and plug in u(0)=0

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0ok. when I factor \(a^2+6a+12=0\) I get \((i a+\sqrt{3}3 i) (i a+\sqrt{3}+3 i)\)

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3that mean it is sinosoid

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0which means it's \((\sqrt{3}\pm(3+a)i)=0\) how do i make it look like the general solution?

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0can you help? because I'm so confused

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3hold on, my computer is apparently running on steam engine

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3a=3 (+) i* sqrt(3) which is \[Ae^{(3+ i \sqrt{3})t}+Be^{(3 i \sqrt{3})t}\]

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3use euler identity and you will get A sin( ) + B cos ()

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0now i can see it's \[Ae^{3}sin(\sqrt{3}t)+Be^{3}cos(\sqrt{3}t)\]then all i have to do is plug in the initial values, right?

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3you are missing t e^(3t)

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0oh yeah. ok there's a t.

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0\[u=e^{−3t}cos\sqrt{3}t+\sqrt{3}e^{−3t}sin\sqrt{3}t\]one more question. how do i determine if this equation is over damped, underdamped, or critically damped?

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3so sin() and cos(x) just go up and down at t go on

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3dw:1362965561866:dw

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3while e(3t) dw:1362965602529:dw

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3just read it http://en.wikipedia.org/wiki/Damping#Critical_damping_.28.CE.B6_.3D_1.29 it will explain much better than I ever can

johnny0929
 2 years ago
Best ResponseYou've already chosen the best response.0oh ok i am reading from my book that all i need to do is compare the coefficient in front of \(u'\) to \(\sqrt{4ac}\) from the equation \(a^2+6a+12=0\)

modphysnoob
 2 years ago
Best ResponseYou've already chosen the best response.3crticle damp goes to zero without oscillating under dampm oscillate
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