Math Is Fun Forum

seiya_001

Member

Registered: 2005-08-23

Posts: 19

differential equation... (i think..)

help....
how do you find the Q from:

d^2q        dq
------ + 8 ----- + 25 q = 300
dt^2         dt

any help would be MOST appreciated...

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"If you can't have more age in your life, then have more life in your age..

kybasche

Member

Registered: 2005-09-22

Posts: 8

Re: differential equation... (i think..)

in my opinion, the easiest way to solve this would be to use laplace transforms.  do you know how to use them?  if not, we can solve it using another method... reply if you want to know more!

~Derek

p.s. Matlab or Maple or a TI-89 calculator can be used to solve differential equations as well if you're not interested in the method

seiya_001

Member

Registered: 2005-08-23

Posts: 19

Re: differential equation... (i think..)

in my opinion, the easiest way to solve this would be to use laplace transforms.  do you know how to use them?

umm..not yet.., what about the other method you mentioned? can it be solved by any diffferential equation? thanx..:)

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"If you can't have more age in your life, then have more life in your age..

kybasche

Member

Registered: 2005-09-22

Posts: 8

Re: differential equation... (i think..)

Alright... I can show how I would solve this w/o using Laplace transforms.  There might be a more streamlined way to do it, but if there is, someone else will have to show you

Here goes...

First we'll create what I think is called the "characteristic polynomial" for the differential equation.  Basically what that means is that we're going to neglect the right hand side (300) for now, and work on the left hand side.  By neglecting the right hand side, we're dealing with something called a homogenous differential equation.

Now, to start, we'll replace all d^n(q)/dt^n with m^n... so the equation becomes

m^2+8m+25 = 0

Now, solve for m, and you get -4 ± 3j (use quadratic formula, or calculator, or whatever...)

What this means is that we've got complex roots of the homogenous equation.  The solution to a differential equation is usually in the form

C*exp(r1*t) + D*exp(r2*t)

where C and D are constants, and r1 and r2 are the roots of the homogenous equation.  Because we have complex roots, we have to apply Euler's formula, which states something like

exp(j*K) = cos(K) + j*sin(K)

You can go through the algebra, but what this ends up boiling down to for our purposes is something that looks like this

C*exp(r*t)*cos(k*t) + D*exp(r*t)*sin(k*t)

where "r" is the real part of the complex root, and "k" is the imaginary part, so right now we've got

q(t)_homogenous = exp(-4*t)*[D*cos(3*t)+E*sin(3*t)]

Now let's deal with the right hand side of the equation that we neglected earlier.  Because it is just a number, there's a bit of a trick that you can use.  Essentially the right hand side is describing the "input" to the system... which means that after the transient behavior of the system dies off (the differential terms) you're left with the "input" and the linear terms of q(t).  What this means in terms of our equation is that

25*q(t) = 300

and if you solve for q(t) you get q(t) = 12, which we will call our "particular" solution... because it is particular to the "input" or something like that.  To get q(t) we'll add the homogenous and particular solutions

Ok, now we're basically done, we've got

q(t) = exp(-4*t)*[D*cos(3*t)+E*sin(3*t)] + 12

Now it's just a matter of plugging in initial conditions (you need (order of differential equation - 1) initial conditions) and solving for the constants, D and E.

If the initial conditions were

q'(0) = 0   and   q(0) = 0

You should end up with

q(t) = exp(-4*t)*[-12*cos(3*t) - 16*sin(3*t)] + 12

I'll spare you the algebra, I'm sure you can do it (assuming you can take the derivative of q(t).

So I think that's it, if something's unclear, which I'm sure there's at least a few things, post back and I'll see if I can try to clear it up.  Enjoy!

~Derek

seiya_001

Member

Registered: 2005-08-23

Posts: 19

Re: differential equation... (i think..)

wow, how do you get all that..!!
thanks a lot dude..i'll need some time to understand it all but i'll try...:)

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"If you can't have more age in your life, then have more life in your age..

kybasche

Member

Registered: 2005-09-22

Posts: 8

Re: differential equation... (i think..)

It's actually not that difficult after a while... for simple linear differential equations (like this one) it's mostly just a matter of following a sort of algorithm.

Find roots of characteristic equation
Determine "class" of roots (complex, distinct real, repeated real)
Form your "base" equation (C*exp(r1*t)+D*exp(r2*t)+etc...) depending on the types of roots
Solve for constants using initial conditions

Have you solved diffeq's before?  If not, I can see how it would seem a bit daunting   They are quite useful, as you could probably imagine.  I'm taking an Automatic Controls class, and what it basically boils down to is using the different properties of differential equations to choose an equation that will model a system to gain a desired output (like maximum value, settling time, etc...)

Anyways, that was a bit off topic.  Like I said, if there's something that's unclear, let me know!

seiya_001

Member

Registered: 2005-08-23

Posts: 19

Re: differential equation... (i think..)

Have you solved diffeq's before?  If not, I can see how it would seem a bit daunting smile

yes, but not with high order equations and all (d^2q/dt^2 or higher)

a question, what if the right hand side of the equation is not constant, let say it's a function also then what would you do with the 'particular solution'? thanks before..(btw in what year are you in college?)

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"If you can't have more age in your life, then have more life in your age..

kybasche

Member

Registered: 2005-09-22

Posts: 8

Re: differential equation... (i think..)

So far as having the RHS not equal to a constant....

There are general methods for solving the equations depending on the type of function on the right hand side.  For instance... if you've got

left hand side = cos(t)

your particular solution would look something like

A*cos(t) + B*sin(t)

I don't know if you have access to a Diff.Eq. book, but there should be a list of the different things that you can end up with depending on the right hand side.  Another common one would be something like

left hand side = A*t^n + ... +B*t + C

then the particular solution would be in the form

D*t^(n-1) + ... + E*t + F

fill in the '...' with the descending powers of t

as you can see, the more complicated the right hand side, the more constants you have to solve for in the end (using the initial conditions).

And college, well... I'm in my senior (4th) year at the University of Michigan majoring in Mechanical Engineering.  Anyone out there looking for an intern for next summer???

~Derek

seiya_001

Member

Registered: 2005-08-23

Posts: 19

Re: differential equation... (i think..)

I don't know if you have access to a Diff.Eq. book, but there should be a list of the different things that you can end up with depending on the right hand side.

any good reference? im' in indonesia so it's rather hard to get a good book with the original language (usually they're translated into indonesian and sometimes it's confusing), but shoot anyway...?

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"If you can't have more age in your life, then have more life in your age..

kybasche

Member

Registered: 2005-09-22

Posts: 8

Re: differential equation... (i think..)

Sorry for the long delay... I haven't got a great reference right off hand.  The book that I used back when I took Diff.Eq. was

A First Course in Differential Equations
The Classic Fifth Edition
Dennis G. Zill
ISBN: 0-534-37388-7
~$40 on Amazon... I'm sure that are cheaper alternatives, but this is the only one that I've used.

~Derek

seiya_001

Member

Registered: 2005-08-23

Posts: 19

Re: differential equation... (i think..)

thanks a lot dude..i'll try to find it in my country.. i'm sure we have it somewhere...:)

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"If you can't have more age in your life, then have more life in your age..