Thread: Differential Equation help!

If any one could help with the following I would be most grateful!

The angular velocity v of a rapidly moving shaft can be shown to obey an equation of the form:

d^3v/dt^3 - d^2v/dt^2 + 2v = 10
(v''' - v'' + 2v = 10)

Under certain conditions (t is the time). Find a particular integral of this equation and hence the general soultion. Then show that except for certain special initial conditions, the angular velocity will vary periodically, with an amplitude increasing with time.


I have started by trying to find the complimentary function as I think that this plus the particular integral eqauls the general soultion. I put the equation in the form k^3 - k^2 + 2 = 0. and Then did a bit of long division to get it in the form (k+1)(k^2-2k+2). Which gives me k= -1, 1 + i, 1- i.

I am not too sure it what I have done is right and also this is where I get a bit stuck. Any help would be great!!!

Originally Posted by goldilocks

If any one could help with the following I would be most grateful!

The angular velocity v of a rapidly moving shaft can be shown to obey an equation of the form:

d^3v/dt^3 - d^2v/dt^2 + 2v = 10
(v''' - v'' + 2v = 10)

Under certain conditions (t is the time). Find a particular integral of this equation and hence the general soultion. Then show that except for certain special initial conditions, the angular velocity will vary periodically, with an amplitude increasing with time.


I have started by trying to find the complimentary function as I think that this plus the particular integral eqauls the general soultion. I put the equation in the form k^3 - k^2 + 2 = 0. and Then did a bit of long division to get it in the form (k+1)(k^2-2k+2). Which gives me k= -1, 1 + i, 1- i.

I am not too sure it what I have done is right and also this is where I get a bit stuck. Any help would be great!!!

So the solution to the homegeneous equation is:

$\displaystyle v_h(t)=Ae^{-t}+Be^{(1+i)t} + C e^{(1-i)t}$,

and a particular integral is:

$\displaystyle v_p(t)=5$

So the general solution is:

$\displaystyle v(t)=v_h(t)+v_p(t)$

RonL

Thanks for your help, it is most appreciated!!! :-)