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#1 2006-05-27 20:08:24

renjer
Member
Registered: 2006-04-29
Posts: 51

Differential Equation

I don't know whether I'm wrong or not, but I get different answers from Scientific Workplace when I do this Differential Equation:

y''+3y'+2y=e^(-t)

Thanks for helping.

View Image: a.jpg

Last edited by renjer (2006-05-27 20:09:42)

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#2 2006-05-27 20:38:47

numen
Member
Registered: 2006-05-03
Posts: 115

Re: Differential Equation

I get the same homogenous answer, so lets see about the rest.

Let (z is a function, z(t))

, so
,

So we get

So


Let z=Ct+D, so z'=C and z''=0; then C must be 1, which gives z=t.

Whohoo, first time using these codes tongue

Last edited by numen (2006-05-27 20:53:53)


Bang postponed. Not big enough. Reboot.

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#3 2006-05-27 21:17:01

renjer
Member
Registered: 2006-04-29
Posts: 51

Re: Differential Equation

The answer seems like the one I had above. Anyway, thanks for showing me another method to solve DEs.

The next question goes: y''+y'+2y=t^2+e^-t+cos t.

Do I just find y''+y'+2y'=cos t? I've already calculated for t^2 and e^-t as was shown by numen. Then after I get all these, I just add them together with A and B. Is it right?

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#4 2006-05-27 22:01:14

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Differential Equation

It's amazing to see someone solving a DE with a "y''" so easily.


X'(y-Xβ)=0

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#5 2006-05-28 01:22:12

renjer
Member
Registered: 2006-04-29
Posts: 51

Re: Differential Equation

Yeah DEs are actually very easy, just that sometimes I do not know the rules.

So anyone know how to answer my original question?

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#6 2006-05-28 02:48:26

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Differential Equation

DEs are easy to solve, when they are given to you by a teacher and made nice.  But most DEs are not only difficult, but impossible, at least with today's methods.  Thats why there is a lot of work in the field of approximating DEs.

Have you learned methods of solving 2nd order DEs?  I would go with Variation of Parameters for this one.  Let me know if you haven't, I can show you how to do it.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2006-05-28 06:31:08

numen
Member
Registered: 2006-05-03
Posts: 115

Re: Differential Equation

Yeah, you can add them all together, you use the superposition principle (I think that's what it's called anyways). DE's are easy if you're being methodic, you rarely encounter higher grade DE's practically, I've been told.

For cosx and sinx I prefer using complex e-functions. Like cosx = Re[e^(ix)]. Try that method, it's really good imho, also if you have cos(7x) or similiar it works great.
I can give you a full solution to it later if you want, I got math exams tomorrow smile

Last edited by numen (2006-05-28 06:32:14)


Bang postponed. Not big enough. Reboot.

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#8 2006-05-30 03:27:00

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Differential Equation

COMPLEX FUNCTION????
faintfaintfaint


X'(y-Xβ)=0

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#9 2006-05-30 12:48:30

renjer
Member
Registered: 2006-04-29
Posts: 51

Re: Differential Equation

I've just learnt another way of solving DEs from lecture notes from another university.

But numen, I've never heard of solving using complex functions.

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#10 2006-05-31 02:55:55

numen
Member
Registered: 2006-05-03
Posts: 115

Re: Differential Equation

I'd be interested in hearing what you learnt smile

If you want to try solving your question with complex methods, use the fact that


If you have, for example, sin(7t), you simply put a 7 in the exponent of e^(ix).

Your question is the most simple example, cos(t), so I think you can follow it quite easily.

---

We know that

from above. Let
and
, so we get:

,

And so

Which means that

Thus, we must have that

since all derivatives disappear, which gives

So

Which becomes

Since

, this must mean that

big_smile

I hope I didn't miss anything out.


Bang postponed. Not big enough. Reboot.

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#11 2006-06-03 01:04:06

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: Differential Equation

That's it!
But how can i learn complex functions and differentials? What kind of books?
I'm eager to learn it,  could you recommand me a book on this kinda stuff? thanks roll

Last edited by George,Y (2006-06-03 21:25:40)


X'(y-Xβ)=0

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