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Question: How to convert expression in hypergeom to integral form?

Question:How to convert expression in hypergeom to integral form?

nm 8552 Maple 2019
convert dsolve int hypergeom odetest duplicate-question
July 26 2019
1

In answers given in 

In https://www.mapleprimes.com/questions/227546-How-To-Make-Odetest-Verify-Dsolve

It shows that odetest() did not verify a solution to ODE becuase solution was using hypergeom special functions. If the solution to the ODE was in integral form, then odetest() will verify it OK.

But what to do if the solution I want to verify is already in hypergoem? If I try odetest() it will fail to verify now. Then I can try to convert the solution to integral form and try again.

But when  using convert(sol,Int) followed by odetest() it did not work.

The solutions I try to verify are hand solutions or book solutions, and not coming from dsolve. 

But some of them are the same solution that comes from dsolve() when not using the useInt option. 

Also, I am doing this all inside a Maple program. It is not an interactive process. So I can't do plots and look at them to decide on anything. So verification must all be implemented in code.

The question is: Why did convert(hand_solution,Int) not give the same result as dsolve(ode,useInt)? Is there another way around this? (May be I am asking for too much in this one based on answers in the above link, So that is OK if not possible. But I really like the solution given when using "useInt" option. Much more clear than otherwise).
 

> 

restart;
> 

ode := diff(y(x), x)*(x^3 + 1)^(2/3) + (1 + y(x)^3)^(2/3) = 0;
sol_int:=dsolve(ode,useInt);
odetest(sol_int,ode); #OK now, since solution in integral form

(diff(y(x), x))*(x^3+1)^(2/3)+(1+y(x)^3)^(2/3) = 0

Int(1/(x^3+1)^(2/3), x)+Intat(1/(_a^3+1)^(2/3), _a = y(x))+_C1 = 0

0

> 

hand_solution:= x*hypergeom([1/3, 2/3], [4/3], -x^3) + y(x)*hypergeom([1/3, 2/3], [4/3], -y(x)^3) + _C1 = 0;
convert(hand_solution,Int); #Why this did not give same result as ABOVE?

x*hypergeom([1/3, 2/3], [4/3], -x^3)+y(x)*hypergeom([1/3, 2/3], [4/3], -y(x)^3)+_C1 = 0

(2/9)*x*Pi*3^(1/2)*(Int(1/(_t1^(1/3)*(1-_t1)^(1/3)*(x^3*_t1+1)^(1/3)), _t1 = 0 .. 1))/GAMMA(2/3)^3+(2/9)*y(x)*Pi*3^(1/2)*(Int(1/(_t1^(1/3)*(1-_t1)^(1/3)*(y(x)^3*_t1+1)^(1/3)), _t1 = 0 .. 1))/GAMMA(2/3)^3+_C1 = 0

> 

odetest(%,ode); #does not give zero

-y(x)^3*(1+y(x)^3)^(2/3)*(Int(_t1^(2/3)/((1-_t1)^(1/3)*(y(x)^3*_t1+1)^(4/3)), _t1 = 0 .. 1))+(x^3+1)^(2/3)*(Int(_t1^(2/3)/((1-_t1)^(1/3)*(x^3*_t1+1)^(4/3)), _t1 = 0 .. 1))*x^3-(x^3+1)^(2/3)*(Int(1/(_t1^(1/3)*(1-_t1)^(1/3)*(x^3*_t1+1)^(1/3)), _t1 = 0 .. 1))+(1+y(x)^3)^(2/3)*(Int(1/(_t1^(1/3)*(1-_t1)^(1/3)*(y(x)^3*_t1+1)^(1/3)), _t1 = 0 .. 1))

> 

 

 

Maple 2019.1

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