available here, I get the solution using Maple V Release 5.1. Eg:

kernelopts(version);

Maple V, Release 5.1, IBM INTEL NT, Nov 05 1998, WIN-5510-980921-1

pdsolve(-a*u(t,x)+diff(u(t,x),t)+b*diff(u(t,x),x,x)=0,u(t,x),build);

                                     sqrt(b _c[1] - b a) x
  u(t, x) = _C1 exp(_c[1] t) _C2 sin(---------------------)
                                               b

                                    sqrt(b _c[1] - b a) x
         + _C1 exp(_c[1] t) _C3 cos(---------------------)
                                              b

I find ?identify a bit cryptic and there is no example of identify with option extension.

I find ?identify a bit cryptic and there is no example of identify with option extension.

at least, I would find it simpler with symbols instead of floats:

f:=(((((X1  * (((X3  / X1 ) + (X2  * ((((((X1  * (((X1  / X1 ) 
+ (X4  + X1 )) / (X4  / ((X2  / X3 ) / (-0.570992561856773))))) 
- (X1  - (-0.5457363063736358))) * X2 ) + X1 ) + X3 ) / X2 ))) 
/ (X4  / ((X4  + X2 ) / (-0.570992561856773))))) - (X1  - 
(-0.5457363063736358))) * X2 ) + X1 ) * (((0.2061949348454455 * 
((-0.6832734709973418 + X3 ) / (-0.570992561856773))) - X1 ) / 
(X4  - 0.2061949348454455))):

g:=subs(zip(`=`,convert(indets(f,float),list),
[seq(a||i,i=1..nops(indets(f,float)))]),f);

((X1*(X3/X1+(X1*(1+X4+X1)*X2*a1/(X4*X3)-X1+a2)*X2+X1+X3)*
(X4*a1+X2*a1)/X4-X1+a2)*X2+X1)*(a5+X3*a3-X1)/(X4+a4)

at least, I would find it simpler with symbols instead of floats:

f:=(((((X1  * (((X3  / X1 ) + (X2  * ((((((X1  * (((X1  / X1 ) 
+ (X4  + X1 )) / (X4  / ((X2  / X3 ) / (-0.570992561856773))))) 
- (X1  - (-0.5457363063736358))) * X2 ) + X1 ) + X3 ) / X2 ))) 
/ (X4  / ((X4  + X2 ) / (-0.570992561856773))))) - (X1  - 
(-0.5457363063736358))) * X2 ) + X1 ) * (((0.2061949348454455 * 
((-0.6832734709973418 + X3 ) / (-0.570992561856773))) - X1 ) / 
(X4  - 0.2061949348454455))):

g:=subs(zip(`=`,convert(indets(f,float),list),
[seq(a||i,i=1..nops(indets(f,float)))]),f);

((X1*(X3/X1+(X1*(1+X4+X1)*X2*a1/(X4*X3)-X1+a2)*X2+X1+X3)*
(X4*a1+X2*a1)/X4-X1+a2)*X2+X1)*(a5+X3*a3-X1)/(X4+a4)

is also proven algebraically?

is also proven algebraically?

is possible by right click on the figure and with both "Element properties" > "Alternate text" and "Title"

is possible by right click on the figure and with both "Element properties" > "Alternate text" and "Title"

I see that the difference in the use of memory space is also significant. Presumably also for this difference in design. I wonder whether this difference has a technical reason or it is just an historical issue.

I see that the difference in the use of memory space is also significant. Presumably also for this difference in design. I wonder whether this difference has a technical reason or it is just an historical issue.

A way that seems to me simple (at least with Maple) is converting to an ODE, more or less along this sketch:

id1:=hypergeom([-1/2,1/4,3/4], [1/3,2/3], 
-(1-2*r)^2/27/r^4/(1-r)^4) = (1-2*r+4*r^3-2*r^4)/6/r^2/(1-r)^2; 
l1:=lhs(id1);
r1:=rhs(id1);
dl1:=PDEtools[dpolyform](f(r)=l1, no_Fn);
#gfun:-holexprtodiffeq(l1,f(r)) could also be used
odetest(f(r)=r1,dl1);
                                 [0]

Is conversion to an ODE the standard way to verify these identities?

A way that seems to me simple (at least with Maple) is converting to an ODE, more or less along this sketch:

id1:=hypergeom([-1/2,1/4,3/4], [1/3,2/3], 
-(1-2*r)^2/27/r^4/(1-r)^4) = (1-2*r+4*r^3-2*r^4)/6/r^2/(1-r)^2; 
l1:=lhs(id1);
r1:=rhs(id1);
dl1:=PDEtools[dpolyform](f(r)=l1, no_Fn);
#gfun:-holexprtodiffeq(l1,f(r)) could also be used
odetest(f(r)=r1,dl1);
                                 [0]

Is conversion to an ODE the standard way to verify these identities?

Typically I get an error message:

The MySQL error was: Too many connections.

Apparently, a problem with access to the database.

Currently I am back with AVG 7.5. If I have time during the weekend I will make some experiments like that with AVG 8.