How do I solve this equation for b1, b2, b3 using Maple or Mathematica?

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August 8th 2010, 04:04 PM
simy

How do I solve this equation for b1, b2, b3 using Maple or Mathematica?

(tH′′)^2 = [H′(2n + t + λ) - H]^2 − 4(tH′ − H + δ)[(H′)^2 + λH′]

where H(t) / (n^2)λ = (summation from k=1 to infinity) of [bk / (t^k)]

((where bk means b with subscript k))

I hope someone can help me..I desperately need to know how to do this!!

Thanks
August 9th 2010, 02:51 AM
Ackbeet

Well, in Mathematica you'd type the following command (I'm assuming this is a differential equation, right?):

DSolve[(t H''[t])^2==(H'[t](2n+t+lambda)-H[t])^2-4(t H'[t]-H[t]+delta)((H'[t])^2+lambda H'[t]),H[t],t]

You should double-check that notation, however, and make sure it's correct against your DE. In particular, I'm not sure whether you meant H'[t](2n+t+lambda) or H'(2n+t+lambda).

I don't think Mathematica is going to be able to handle that. At least, my copy of Mathematica 4 can't handle it. Maybe more recent versions have more powerful solvers. I'm talking here about exact, closed-form solutions. Mathematica can probably get you a numerical solution the same as MATLAB or any other solver routine. Do you need a numerical solution?
August 9th 2010, 06:56 AM
simy

Thanks

I meant H'[t](2n+t+lambda).

Also, I need solutions in terms of n and lambda (except for the first value which my lecturer told me will be b1=1).

I tried entering eval and then the above expression in Maple but it just kept giving me the same expression again and again.
August 9th 2010, 09:35 AM
Ackbeet

Hang on a second. Let me rephrase the question. You're asked to solve

$(t H''(t,n,\lambda))^{2}=[H'(t,n,\lambda)(2n+t+\lambda)-H(t,n,\lambda)]^{2}$
$-4(t H'(t,n,\lambda)-H(t,n,\lambda)+\delta)[(H'(t,n,\lambda))^{2}+\lambda H'(t,n,\lambda)],$

where

$\displaystyle{H(t,n,\lambda)=\lambda n^{2}\sum_{k=1}^{\infty}\frac{b_{k}}{t^{k}},}$

and differentiation is with respect to $t$ . Is that correct? If so, I think I would differentiate your expression for $H$ as required, and convert the DE into a difference equation. Do you have any initial conditions?

So, the first step is differentiating:

$\displaystyle{H'(t,n,\lambda)=\lambda n^{2}\sum_{k=1}^{\infty}\frac{(-k) b_{k}}{t^{k+1}},}$ and

$\displaystyle{H''(t,n,\lambda)=\lambda n^{2}\sum_{k=1}^{\infty}\frac{(-k)(-k-1) b_{k}}{t^{k+2}}.}$

Then you could plug that into your DE. You'd get some messy series multiplication going on there, but it might be doable. You'd get a recurrence relation for the $b_{k}$ 's.
August 9th 2010, 11:45 AM
simy

That's right. That's exactly what I have to solve, but my lecturer wants me to do it using Maple and not by hand and since I've never used it before, I can't figure out what commands to enter.
August 9th 2010, 11:53 AM
Ackbeet

I haven't a clue on how to use Maple to solve that. With Mathematica, I'd probably play around with the Series command, and the RSolve command, to simplify your work. Look up the help on those commands, if you decide on Mathematica.

But first, you need to get your DE looking like an actual recurrence relation. I'd work with that first, referencing the usual series solution method, which you should be able to adapt to your DE, as nasty as it is.