I'd like to solve this ODE

But not sure how to write it in Maple. In Mathematica, this is what I do

ClearAll[F, x, y, t];
ode = D[F[x[t], y[t]], x[t]]*D[x[t], t] + D[F[x[t], y[t]], y[t]]*D[y[t], t] == 0;
DSolve[ode, F[x[t], y[t]], {x[t], y[t]}]

and it gives

In Maple, I tried

ode:=diff(F(x(t),y(t)),x(t))*diff(x(t),t)+diff(F(x(t),y(t)),y(t))*diff(y(t),t)=0

Error, invalid input: diff received x(t), which is not valid for its 2nd argument

ode:=D[1](F)(x(t),y(t))*diff(x(t),t)+D[2](F)(x(t),y(t))*diff(y(t),t)=0;
dsolve(ode,F(x(t),y(t)))

Error, (in dsolve) expected the indeterminate function as, say, F(x) where x is of type "name" - and also cannot be a procedure name. Received: [F(x(t), y(t))]

What is the correct way to write this in Maple?

Do you agree this solution given by Maple is not correct?

restart;
pde := diff(u(x,t),t)+diff(u(x,t),x)=0;
bc  := D[1](u)(0,t)=0;
ic  := u(x,0)=exp(-x^2);
sol:=pdsolve([pde,ic,bc],u(x,t)) assuming x>0,t>0;
pdetest(sol,pde)

Result of pdetest should be zero.

I think the PDE itself is not well posed (I copied it from different forum to see what Maple does with it). But still the solution clearly does not satisfy the PDE itself for x not zero. 

Maple 2019.2.1 with Physics version 573

Hello;

Maple 2019.2.1 on windows 10

Is there a trick to make inttrans:-fourier(erf(x),x,k) return the Fourier transform of the error function? Now, using Maple 2019.2.1 it returns unevaluated. But direct application of Fourier transform integral does return the correct result. So why inttrans does not work? 
 

Download erf.mw

What is the correct way to specify region where solution of a PDE is needed? For example, I am trying to verify my hand solution to this HW problem: Solve Poisson PDE in 2D 

The above is the only information given in the textbook. So it is only in the upper half plane. If I type this

interface(showassumed=0);
pde := VectorCalculus:-Laplacian(u(x,y),[x,y])=-1/(1+y);
bc:=u(x,0)=0;
pdsolve([pde,bc],u(x,y)) assuming y>0

Maple gives 

But I get by hand using method of images is

So not exactly the same. I think I made mistake in my solution. But I am also not sure that just saying "assuming y>0" is doing what I think it is supposed to do. For example, suppose we want to solve the same PDE say in the first quadrant. Typing

pdsolve([pde,bc],u(x,y)) assuming y>0,x>0

Gives same solution. But the solution should be different. And typing

pdsolve([pde,bc],u(x,y)) 

Gives same answer as well.

So I think I need another way to tell Maple the region of the solution. i.e. I need to tell Maple to use the Laplacian for the upper half plane only and not the Laplacian in the whole 2D space.

Any suggestions what to do and how to handle such problems?

Thank you

Hello;

What trick if any is needed to obtain zero for this sum in Maple?

sum(sin(Pi*n/2)*sin(n*Pi*(x + 1)/2)*cos(n*Pi*t/2),n=1..infinity)

The above sum, according to Mathematica is zero. I am trying to see if same result can be obtained by Maple in order to verify this result. It is possible ofcourse that Mathematica result is not correct. I am also trying to verify the sum is zero by hand, but no success so far.

Here is Mathematica result

Using Maple 2019.2 on windows 10. 

Thanks

Download q2.mw