@C_R 

The problem in Maple is: '"automatic simplification' and 'evaluation'".

See here , here and here.

Cross post here. and here

1.What you need PDEtools for that?

2.You want to solve a Nonlinear Volterra Integro Differential Equation ?, if yes post the equation.

I'm voting to close this question as off-topic because there is no well-posed question in this post; the OP is simply asking for somebody to act as a free coding service.

I'm voting to close this question as off-topic because there is no well-posed question in this post; the OP is simply asking for somebody to act as a free coding service.

This line of code dosen't work:

sum(sum((k[1]+3)!*((-1)^(k-k[1]-1)/2)*Y[k[1]+3]/(k[1]!*(k-k[1])!,k[1]=0..k)

?

@Axel Vogt 

I don't have reference.

I'm played with this sum a few minutes and I found  generalization.

restart;

n := 2;

Sum(Beta(k, c)/(a*k + b)^n, k = 1 .. infinity) = GAMMA(c)*hypergeom([1, 1, seq(1 + b/a, j = 1 .. n)], [seq(2 + b/a, j = 1 .. n), 1 + c], 1)/(GAMMA(1 + c)*(a + b)^n);

@escorpsy 

(sum((-1)^k*z^(1 + 2*k)*GAMMA(1/2 + k)/((1 + 2*k)*sqrt(Pi)*GAMMA(1 + k)), k = 0 .. infinity, formal) assuming (0 < z));

#arcsinh(z)

(sum((-1)^k*sqrt(Pi)*z^(2 + 2*k)*GAMMA(1 + k)/(2*(1 + k)*GAMMA(3/2 + k)), k = 0 .. infinity, formal) assuming (0 < z));

#arcsinh(z)^2

And what is the question ?

@Kitonum 

My solution uses 1000 terms,not a few.

If Maple nothing returns,it means that it does not know the answer.

@acer 

Thanks for workaround for Maple 2021.2 and Maple 18.02.

@computermemory 

Probably Yes,We can speculate because I don't know the solution.

@computermemory 

convert(solution,trig);

convert(solution,sin);

Reagards.

I don't see any differential equations ?Please post a maple code?