## How to solve the given BVP using differential tran...

HI, I have numerically solved the given problem using the dsolve command But I want to solve the same problem using the Differential transformation method.
Can anyone help me to get the series solution for the given problem using DTM.

I want to compare the numerical results with DTM results when lambda =0.5.

eqn1 := diff(f(eta), `\$`(eta, 3))+f(eta)*(diff(f(eta), `\$`(eta, 2)))-(diff(f(eta), eta))^2-lambda*(diff(f(eta), eta)) = 0.

eqn2 := diff(theta(eta), `\$`(eta, 2))+f(eta)*(diff(theta(eta), eta))*Pr = 0

Bcs := (D(f))(0) = 1, f(0) = 0, (D(f))(infinity) = 0, theta(0) = 1, theta(infinity) = 0;

[lambda = .5, Pr = 6.3]

## package of Fractional Symmetry ASPv4.6.3.mpl...

restart;

DESOLVII_V5R5 (March 2011)(c), by Dr. K. T. Vu, Dr. J.

Carminati and Miss G. Jefferson

The authors kindly request that this software be referenced, if

it is used in work eventuating in a publication, by citing

the article:

K.T. Vu, G.F. Jefferson, J. Carminati, Finding generalised

symmetries of differential equations

using the MAPLE package DESOLVII,Comput. Phys. Commun. 183

(2012) 1044-1054.

-------------

ASP (November 2011), by Miss G. Jefferson and Dr. J. Carminati

The authors kindly request that this software be referenced, if

it is used in work eventuating in a publication, by citing

the article:

G.F. Jefferson, J. Carminati, ASP: Automated Symbolic

Computation of Approximate Symmetries

of Differential Equations, Comput. Phys. Comm. 184 (2013)

1045-1063.

[classify, comtab, defeqn, deteq_split, extgenerator, gendef,

genvec, icde_cons, liesolve, mod_eq, originalVar, pdesolv,

reduceVar, reduceVargen, symmetry, varchange]

ASP := _m2229977204928

with(ASP);
[ApproximateSymmetry, applygenerator, commutator]

with(desolv);
[classify, comtab, defeqn, deteq_split, extgenerator, gendef,

genvec, icde_cons, liesolve, mod_eq, originalVar, pdesolv,

reduceVar, reduceVargen, symmetry, varchange]

FracSym (April 2013), by Miss G. Jefferson and Dr. J. Carminati

The authors kindly request that this software be referenced, if

it is used in work eventuating in a publication, by citing:

G.F. Jefferson, J. Carminati, FracSym: Automated symbolic

computation of Lie symmetries

of fractional differential equations, Comput. Phys. Comm.

Submitted May 2013.

with(FracSym);
[Rfracdiff, TotalD, applyFracgen, evalTotalD, expandsum, fracDet,

fracGen, split]

Rfracdiff(u(x, t), t, alpha);
alpha
D[t     ](u(x, t))

Rfracdiff(u(x, t) &* v(x, t), t, alpha);
infinity
-----
\
)                          (alpha - n)              n
/     binomial(alpha, n) D[t           ](u(x, t)) D[t ](v(x, t))
-----
n = 0

Rfracdiff(v(x, t) &* u(x, t), t, alpha);
infinity
-----
\
)                          (alpha - n)              n
/     binomial(alpha, n) D[t           ](v(x, t)) D[t ](u(x, t))
-----
n = 0

Rfracdiff(u(x, t) &* v(x, t), t, 2);
/  2         \
| d          |             / d         \ / d         \
|---- u(x, t)| v(x, t) + 2 |--- u(x, t)| |--- v(x, t)|
|   2        |             \ dt        / \ dt        /
\ dt         /

/  2         \
| d          |
+ u(x, t) |---- v(x, t)|
|   2        |
\ dt         /

TotalD(xi[x](x, y), x, 2);
2
D[x ](xi[x](x, y))

evalTotalD([%], [y], [x]);
[     /  2             \     /   2              \
[   2 | d              |     |  d               |
[y_x  |---- xi[x](x, y)| + 2 |------ xi[x](x, y)| y_x
[     |   2            |     \ dy dx            /
[     \ dy             /

/  2             \]
/ d             \   | d              |]
+ y_xx |--- xi[x](x, y)| + |---- xi[x](x, y)|]
\ dy            /   |   2            |]
\ dx             /]

fde1 := Rfracdiff(u(x, t), t, alpha) = -u(x, t)*diff(u(x, t), x) - diff(u(x, t), x, x) - diff(u(x, t), x, x, x) - diff(u(x, t), x, x, x, x);
alpha                      / d         \
fde1 := D[t     ](u(x, t)) = -u(x, t) |--- u(x, t)|
\ dx        /

/  2         \   /  3         \   /  4         \
| d          |   | d          |   | d          |
- |---- u(x, t)| - |---- u(x, t)| - |---- u(x, t)|
|   2        |   |   3        |   |   4        |
\ dx         /   \ dx         /   \ dx         /

deteqs := fracDet([fde1], [u], [x, t], 2);
Intervals/values considered for the fractional derivative/s:

{0 < alpha, alpha < 1}

[
[
[[  2
[[ d                     d
deteqs := [[---- eta[u](x, t, u), --- xi[t](x, t, u),
[[   2                   du
[[ du

d                   d                   d
--- xi[t](x, t, u), --- xi[t](x, t, u), --- xi[x](x, t, u),
du                  dx                  du

2                    2
d                   d                    d
--- xi[x](x, t, u), ---- xi[t](x, t, u), ---- xi[t](x, t, u),
du                    2                    2
du                   du

2
d                         / d                \
---- xi[t](x, t, u), alpha |--- xi[x](x, t, u)|,
2                       \ dt               /
du

2
/ d                \   d
alpha |--- xi[x](x, t, u)|, ---- xi[t](x, t, u),
\ du               /     2
du

2                    2                    3
d                    d                    d
---- xi[x](x, t, u), ---- xi[x](x, t, u), ---- xi[t](x, t, u),
2                    2                    3
du                   du                   du

3                    3                    4
d                    d                    d
---- xi[t](x, t, u), ---- xi[x](x, t, u), ---- xi[t](x, t, u),
3                    3                    4
du                   du                   du

4
d
---- xi[x](x, t, u),
4
du

/  2                \
| d                 |     / d                \
-6 |---- xi[t](x, t, u)| - 3 |--- xi[t](x, t, u)|,
|   2               |     \ dx               /
\ dx                /

/ d                \     / d                \
alpha |--- xi[t](x, t, u)| - 4 |--- xi[x](x, t, u)|,
\ dt               /     \ dx               /

/ d                \
|--- xi[t](x, t, u)| (alpha - 1),
\ du               /

/   2                 \
/ d                \      |  d                  |
-3 |--- xi[t](x, t, u)| - 12 |------ xi[t](x, t, u)|,
\ du               /      \ dx du               /

/ d                \
alpha |--- xi[t](x, t, u)| (alpha - 1),
\ du               /

/ d                \
alpha |--- xi[x](x, t, u)| (alpha - 1),
\ du               /

/  2                \      /   3                  \
| d                 |      |  d                   |
-3 |---- xi[t](x, t, u)| - 12 |------- xi[t](x, t, u)|,
|   2               |      |      2               |
\ du                /      \ dx du                /

/   2                 \
|  d                  |
alpha |------ xi[t](x, t, u)| (alpha - 1),
\ du dt               /

/   2                 \
|  d                  |
alpha |------ xi[x](x, t, u)| (alpha - 1),
\ du dt               /

/  2                \
| d                 |
alpha |---- xi[t](x, t, u)| (alpha - 1),
|   2               |
\ du                /

/  2                \
| d                 |
alpha |---- xi[x](x, t, u)| (alpha - 1),
|   2               |
\ dt                /

/  2                \
| d                 |
alpha |---- xi[x](x, t, u)| (alpha - 1),
|   2               |
\ du                /

/  3                \     /   4                  \
| d                 |     |  d                   |
-|---- xi[t](x, t, u)| - 4 |------- xi[t](x, t, u)|,
|   3               |     |      3               |
\ du                /     \ dx du                /
/   2                 \
/ d                \     |  d                  |
-|--- xi[t](x, t, u)| - 4 |------ xi[t](x, t, u)|
\ du               /     \ dx du               /

/ d                \     / d                \
+ alpha |--- xi[t](x, t, u)|, -4 |--- xi[x](x, t, u)|
\ du               /     \ du               /

/  2                 \      /   2                 \
| d                  |      |  d                  |
+ 4 |---- eta[u](x, t, u)| - 16 |------ xi[x](x, t, u)|,
|   2                |      \ dx du               /
\ du                 /
/  2                 \
/ d                \     | d                  |
-3 |--- xi[x](x, t, u)| + 3 |---- eta[u](x, t, u)|
\ du               /     |   2                |
\ du                 /

/   2                 \     /  3                \
|  d                  |     | d                 |
- 12 |------ xi[x](x, t, u)|, -4 |---- xi[t](x, t, u)|
\ dx du               /     |   3               |
\ dx                /

/  2                \
/ d                \     | d                 |
- 2 |--- xi[t](x, t, u)| - 3 |---- xi[t](x, t, u)|,
\ dx               /     |   2               |
\ dx                /
/   2                 \      /   3                  \
|  d                  |      |  d                   |
-6 |------ xi[t](x, t, u)| - 12 |------- xi[t](x, t, u)|
\ dx du               /      |   2                  |
\ dx  du               /

/ d                \
- 2 |--- xi[t](x, t, u)|,
\ du               /

/ d                \
alpha |--- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
\ du               /
/  2                \     /  3                 \
| d                 |     | d                  |
-6 |---- xi[x](x, t, u)| + 6 |---- eta[u](x, t, u)|
|   2               |     |   3                |
\ du                /     \ du                 /

/   3                  \     /   3                  \
|  d                   |     |  d                   |
- 24 |------- xi[x](x, t, u)|, -3 |------- xi[t](x, t, u)|
|      2               |     |      2               |
\ dx du                /     \ dx du                /

/    4                  \   /  2                \
|   d                   |   | d                 |        /
- 6 |-------- xi[t](x, t, u)| - |---- xi[t](x, t, u)|, alpha |
|   2   2               |   |   2               |        \
\ dx  du                /   \ du                /

d                \     / d                \
--- xi[t](x, t, u)| - 3 |--- xi[x](x, t, u)|
dt               /     \ dx               /

/   2                  \     /  2                \
|  d                   |     | d                 |
+ 4 |------ eta[u](x, t, u)| - 6 |---- xi[x](x, t, u)|,
\ dx du                /     |   2               |
\ dx                /

/   2                 \
|  d                  |
alpha |------ xi[t](x, t, u)| (alpha - 1) (alpha - 2),
\ du dt               /

/  2                \
| d                 |
alpha |---- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
|   2               |
\ du                /
/   2                 \     /   3                  \
|  d                  |     |  d                   |
-3 |------ xi[t](x, t, u)| - 6 |------- xi[t](x, t, u)|
\ dx du               /     |   2                  |
\ dx  du               /

/
/ d                \         / d                \        |
- |--- xi[t](x, t, u)| + alpha |--- xi[t](x, t, u)|, alpha |
\ du               /         \ du               /        |
\
/  2                \     /   2                  \
| d                 |     |  d                   |
-alpha |---- xi[t](x, t, u)| + 2 |------ eta[u](x, t, u)|
|   2               |     \ du dt                /
\ dt                /

/  2                \\   /  3                \
| d                 ||   | d                 |
+ |---- xi[t](x, t, u)||, -|---- xi[x](x, t, u)|
|   2               ||   |   3               |
\ dt                //   \ du                /

/   4                  \   /  4                 \
|  d                   |   | d                  |
- 4 |------- xi[x](x, t, u)| + |---- eta[u](x, t, u)|,
|      3               |   |   4                |
\ dx du                /   \ du                 /
/  2                \
/ d                \   | d                 |
-u |--- xi[t](x, t, u)| - |---- xi[t](x, t, u)|
\ dx               /   |   2               |
\ dx                /

/  3                \   /  4                \
| d                 |   | d                 |
- |---- xi[t](x, t, u)| - |---- xi[t](x, t, u)|,
|   3               |   |   4               |
\ dx                /   \ dx                /

/   3                  \
|  d                   |
alpha |------- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
|      2               |
\ du dt                /

/   3                  \
|  d                   |
alpha |------- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
|   2                  |
\ du  dt               /

/  3                \
| d                 |
alpha |---- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
|   3               |
\ du                /
/  2                 \
/ d                \     | d                  |
-3 |--- xi[x](x, t, u)| + 3 |---- eta[u](x, t, u)|
\ du               /     |   2                |
\ du                 /

/   2                 \      /   3                   \
|  d                  |      |  d                    |
- 9 |------ xi[x](x, t, u)| + 12 |------- eta[u](x, t, u)|
\ dx du               /      |      2                |
\ dx du                 /

/   3                  \
|  d                   |        / d                \
- 18 |------- xi[x](x, t, u)|, alpha |--- xi[t](x, t, u)| u
|   2                  |        \ du               /
\ dx  du               /

/   3                  \     /   4                  \
|  d                   |     |  d                   |
- 3 |------- xi[t](x, t, u)| - 4 |------- xi[t](x, t, u)|
|   2                  |     |   3                  |
\ dx  du               /     \ dx  du               /

/   2                 \
|  d                  |   / d                \          /
- 2 |------ xi[t](x, t, u)| - |--- xi[t](x, t, u)| u, alpha |
\ dx du               /   \ du               /          \

/  3                \
d                \     | d                 |
--- xi[t](x, t, u)| - 4 |---- xi[x](x, t, u)|
dt               /     |   3               |
\ dx                /

/  2                \
| d                 |     / d                \
- 3 |---- xi[x](x, t, u)| - 2 |--- xi[x](x, t, u)|
|   2               |     \ dx               /
\ dx                /

/   3                   \     /   2                  \
|  d                    |     |  d                   |
+ 6 |------- eta[u](x, t, u)| + 3 |------ eta[u](x, t, u)|,
|   2                   |     \ dx du                /
\ dx  du                /
/  2                \   /  3                 \
| d                 |   | d                  |
-|---- xi[x](x, t, u)| + |---- eta[u](x, t, u)|
|   2               |   |   3                |
\ du                /   \ du                 /

/   3                  \     /   4                   \
|  d                   |     |  d                    |
- 3 |------- xi[x](x, t, u)| + 4 |------- eta[u](x, t, u)|
|      2               |     |      3                |
\ dx du                /     \ dx du                 /

/    4                  \              /
|   d                   |              |
- 6 |-------- xi[x](x, t, u)|, (alpha - 1) |
|   2   2               |              |
\ dx  du                /              \
/  3                \     /   3                   \
| d                 |     |  d                    |
-alpha |---- xi[t](x, t, u)| + 3 |------- eta[u](x, t, u)|
|   3               |     |      2                |
\ dt                /     \ du dt                 /

/  3                \\
| d                 ||           / d                \
+ 2 |---- xi[t](x, t, u)|| alpha, -u |--- xi[x](x, t, u)|
|   3               ||           \ du               /
\ dt                //

/  2                 \     /   2                 \
| d                  |     |  d                  |
+ |---- eta[u](x, t, u)| - 2 |------ xi[x](x, t, u)|
|   2                |     \ dx du               /
\ du                 /

/   3                   \     /   3                  \
|  d                    |     |  d                   |
+ 3 |------- eta[u](x, t, u)| - 3 |------- xi[x](x, t, u)|
|      2                |     |   2                  |
\ dx du                 /     \ dx  du               /

/   4                  \     /    4                   \
|  d                   |     |   d                    |
- 4 |------- xi[x](x, t, u)| + 6 |-------- eta[u](x, t, u)|,
|   3                  |     |   2   2                |
\ dx  du               /     \ dx  du                 /
/ d                \
-u |--- xi[x](x, t, u)| + eta[u](x, t, u)
\ dx               /

/   2                  \
/ d                \       |  d                   |
+ alpha |--- xi[t](x, t, u)| u + 2 |------ eta[u](x, t, u)|
\ dt               /       \ dx du                /

/  2                \     /   3                   \
| d                 |     |  d                    |
- |---- xi[x](x, t, u)| + 3 |------- eta[u](x, t, u)|
|   2               |     |   2                   |
\ dx                /     \ dx  du                /

/  3                \     /   4                   \
| d                 |     |  d                    |
- |---- xi[x](x, t, u)| + 4 |------- eta[u](x, t, u)|
|   3               |     |   3                   |
\ dx                /     \ dx  du                /

[
[
/  4                \]  [
| d                 |]  [
- |---- xi[x](x, t, u)|], [xi[t](x, 0, u) = 0, (Diff(
|   4               |]  [
\ dx                /]  [

/ d                 \
eta[u](x, t, u), t \$ alpha)) + u |--- eta[u](x, t, u)|
\ dx                /

/    / d                            \\
- u |Diff|--- eta[u](x, t, u), t \$ alpha||
\    \ du                           //

/  3                 \   /  4                 \
| d                  |   | d                  |
+ |---- eta[u](x, t, u)| + |---- eta[u](x, t, u)|
|   3                |   |   4                |
\ dx                 /   \ dx                 /

/infinity
| -----
/  2                 \  |  \
| d                  |  |   )    /    1   /
+ |---- eta[u](x, t, u)|, |  /     |- ----- |binomial(alpha, n)
|   2                |  | -----  \  n + 1 \
\ dx                 /  \ n = 3

/   (alpha - n)              (n + 1)
|D[t           ](u(x, t)) D[t       ](xi[t](x, t, u)) alpha
\

(alpha - n)              (n + 1)
- D[t           ](u(x, t)) D[t       ](xi[t](x, t, u)) n

(alpha - n) / d         \    n
+ D[t           ]|--- u(x, t)| D[t ](xi[x](x, t, u)) n
\ dx        /

\   /Sum(
|   |
|   |
(alpha - n) / d         \    n                 \\\|   |
+ D[t           ]|--- u(x, t)| D[t ](xi[x](x, t, u))|||| + |
\ dx        /                      ///|   |
/   \

/    / d                        \\
binomial(alpha, n) |Diff|--- eta[u](x, t, u), t \$ n||
\    \ du                       //

(alpha - n) (u(x, t)), n = 3 .. infinity)\]
D[t           ]                             |]
|]
|]
|],
|]
/]

]
]
]
]
[xi[x](x, t, u), xi[t](x, t, u), eta[u](x, t, u)], [x, t, u]]
]
]

sol1 := pdesolv(expand(deteqs[1]), deteqs[3], deteqs[4]);
Error, (in desolv/lderivx) cannot determine if this expression is true or false: 1 < x |C:/Program Files/Maple 2020/lib/ASP v4.6.3.txt:4312|

## Maple and Matlab2022a...

1. I use both Maple and Matlab
2. I also install (a stripped down version of) Maple as the "symbolic toolbox" for Matlab using the executable MapleToolbox2022.0WindowsX64Installer.exe, which lives in C:\Program Files\Maple 2022. This gives me acces to (some) symbolic computation capability from within Matlab.
3. This installation process has been working for as long as I remember, certainly more than 10 years
4. With Maple 2022 and Matlab R2022a, this installation process ran with no problems and I can perform symbolic computation within Matlab
5. However, although the Matlab help lists the Maple toolbox as supplemental software (as in all previous releases), I can no longer acces help for Maple from within Matlab - I just get a "Page not found" message
6. The relevant Maple "help" is at the same place within the Matlab folder structure which is C:\Program Files\MATLAB\R2022a\toolbox\maple\html
7. I have just spoken to support at Matlab and they claim tha this must be a Maple (or Maple toolbox installer issue) - so nothing to do with them!
8. Has anyone else had a similar problem andd found a workaround?

## Arial makes sign disappear in MathContainer...

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Vorzeichen.mw

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Does maple software support Benders decomposition technique for Mixed Integer Linear Programming? If no, how we can implement it in maple? Any suggestions.

Thank you

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Anyone experience a delay in typing as the screen fills with text/math etc.?

I'm using Maple 2021 and as the space is filled typing slows. I can finish typing and watch the last 4 keys enter on the screen.

## I have a problem with the "Diff" command....

Hi !

Looks like there is a bug in the inert "Diff" command.

I have Maple 2018 on Windows 10 ,64 bits.

Does Maple consider Diff(f(x),x) to be equal to Diff(f(x),[x]) ?

It should be the same.

Maple displays  that it is equal but keeps in memory something else.

In the attached file, I give a very simple example.

I don't like to say this but my old version of Maple V Release V (1997) is more consistent i.e.

this version shows it's different and  keeps in memory that difference.

diff-problem.mw

I wonder if newer versions have this problem ?

Best regards !

## How does fsolve work?...

When using the built-in fsolve function to find the roots of a polynomial, how does exponentiation occur? For example, x3 is found​​​​​​first, and then to find x4, will he start again from the beginning, that is, x*x*x*x, or will he take the value of x3​​​​​​ and multiply by x? The teacher is interested in finding out this, but I don't know how to find out myself.

## Solve command Issue due to Computational Cost...

Hello Everyone;

Hope you are fine. I am solving system of odes using rk-4 method. For this purpose I formulate the "residual" (on maple file) which is further function of "x" and "y". With the help of discritization point further I convert "residual" into system of ode's. Then i used "sys111 := solve(odes_Combine, `~`[diff](var, t))" to simplify the system. Finnally i applied RK-1. Code is pasted and attached. This all process is for "N=4". When i increase the value of "N", number of Odes increase accordingly. With increasing value of "N" the comand "sys111 := solve(odes_Combine, `~`[diff](var, t))" taking a lot of time due to heavy computation. Is that any way to proceed without this comand for rk-1?

Question1.mw

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## Maxima of the given equation...

Let say,

A= A1+A2+.....................+An

B=B1+B2+.....................+Bn,

C=C1+C2+.....................+Cn

And all the values of A1 to Cn may be both positive or Negative.

Then, how to program to find the Maximum Value of  (A^2+B^2+C^2+A.B+B.C+C.A)^(1/2).

## What can block the import of data from Excel?...

I have a student who then she uses tools/assistent/import data and then the file then Maple claims the Excel file is empty? She uses Maple 2021.2. File works on other my and other student computers.

## why expanding PDE makes Maple hangs?...

To Maple support:

I was investigating this pde from a different forum.

I noticed that when using an expanded version of the pde, Maple hangs. Without expanding the PDE, Maple gives an answer in 2 seconds.

Why does expanding the PDE makes a difference? I do not have an earlier version of Maple on my new PC to check if this is a new issue or not.

 > interface(version);

 > Physics:-Version()

 > restart; pde1:=VectorCalculus:-Laplacian(u(r,theta),'polar'[r,theta]); pde1_expanded:=expand(pde1); bc  := u(1,theta)=sin(theta)^4,u(3,theta)=1; pdsolve([pde1=0,bc],u(r,theta))

 > pdsolve([pde1_expanded=0,bc],u(r,theta)); #HANGS, Waited more than 40 minutes.
 >

## Maple Matrix Question...

Let
"a"

and
"b"

be real numbers and

"A = Matrix(3, 3, [[a, a - 1, -b], [a - 1, a, -b], [b, b, 2*a -

1]])"

,
"B="

"Matrix(5, 5, [[0, a, 3, 0, a], [3, 0, 0, b, 0], [0, 1, b, 0,

1], [b, 0, 0, 1, 0], [0, a, 1, 0, b]])"

(a) Show that if
"0 <= a"

"a <= 1"

and
"b^2 = 2*a*(1 - a)"

, then A is an orthogonal matrix with determinant equal to one.
(b) For what values of a and b is the matrix B singular? Determine the inverse of B (for those values of a and b for which B is invertible).

## Find all rational function solutions to the Kadomt...

Find all rational function solutions to the Kadomtsev-Petviashvili equation

(&PartialD;)/(&PartialD;x);

diff(u, t) + 6*u*diff(u, x) + diff(u, x, x, x) - diff(u, y, y) = 0;

by u = 2
diff(ln, x, x)*f;

=(2 (((&PartialD;)^2)/(&PartialD;x^2) f) f-2 ((&PartialD;)/(&PartialD;x) f)^2)/(f^2);
with
f;
=
(a[1 ]x+a[2] y+a[3] t+a[4])^2+(a[5] x+ a[6] y+a[7] t+a[8])^2+a[9], ;
where
a[i], i=1..9, ;
are real constants.

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