@zhuxian 

I answer by myself.

It will be right if we use the command

'expand(G+u[1,1]*G)'

It is strange, and I don't know the reason.

Maple is silly.

@Carl Love
 

Download ques.mw

The second to last command is normal, but the last one is wrong.
 

@Christian Wolinski 

What do you mean '1D'?

The expression is obtained from

1. compute the EulerLagrange equation of L(t,x,u[],......);

2. some other computations, like addition and differential.

I just copied the expression from output of Maple.

@Carl Love 

Thanks for your help.

I changed a computer and it is normal. I don't know what is the matter with my computer.

Anyway, thanks for your work.

@ecterrab 

Thanks for your help!

I have tried this command, and it works.

However, there are too many solutions, and I want to select some interesting solutions.

There are too many constants (>30) in solution, and they are the form _C1, _C2, _C3, ..., _C35.

I want to set all but one to zero, how to do that?

Thanks for your time.

@ecterrab 

I learn much from your answer. Thanks!

The command "infolevel" is good.

However, how can I get the general solution?

I think maple has find the general solution, but it didn't stop, and continued to build some special solution.

How can I stop it, or other ideas to get the general solution?

For the file I upload, can you help me to get the  results?

Thank you very much. 

@tomleslie 

I forgot plus _C1 to the real_sol, since you can see that k starts from 2 in the constant C_k!

Thanks for that.

However, what I want is the general solutions of the differential equations through maple.

As you can see, the maple-solution is just a special solution.

That A(x,t) satisfies a linear differential equaiton is enough, not a special A(x,t)=C1exp(c1x)C2exp(t/C1).

How can I get the general solution, not by hand?

I will meet some other similiar problems,  I don't want to compute by hand each time, and actually I even don't trust the command 'pdsolve', for that there may be some solutions are hidden!

@ecterrab 

http://www.mapleprimes.com/questions/219023-A-Problem-About-The-Infinitesimals-In-Pdetools

any advice or commence?

thanks!

@ecterrab 

http://www.mapleprimes.com/questions/219023-A-Problem-About-The-Infinitesimals-In-Pdetools

any advice or commence?

thanks!

@Markiyan Hirnyk 

the result i get:

{_eta[u](x, t, u) = _C1*u+_C2, _xi[t](x, t, u) = _xi[t](x, t, u), _xi[x](x, t, u) = 2*(Int(diff(_xi[t](x, t, u), u), u))+(1/2)*(Int(-4*(Int(diff(_xi[t](x, t, u), x, u), u))-_C1+4*(diff(_xi[t](x, t, u), x)), x))+Int(3*_C1+_C2-2*(Int(-(Int(diff(_xi[t](x, t, u), x, u, t), u))+diff(_xi[t](x, t, u), x, t), x))-2*(Int(diff(_xi[t](x, t, u), u, t), u))+2*(diff(_xi[t](x, t, u), t)), t)+_C3}

window10 and maple 18.02

i decide to update maple.

i solved the system by hand, and it is not too complicated.

i don't know why infinite integrals appear in the results from maple.

maybe we shouldn't rely on the computer too much.

@ecterrab 

thank you for your comprehensive answer!

how to solve the system consist of the coefficients?