finding out if expr has the form F(y/x^a)

NB from
http://www.math.utexas.edu/pipermail/maxima/2012/031063.html
write:

Dear all,
I would like to find out if an expression is of (or can be written in) the
general form (y/x)*f(y/x^a), with 'a' a nonzero constant. The difficulty I
have here is the generality of the function f. You can have:
expr:y^3+1/x^(3/2);
or:
expr:(y/x)*sin(y^2/x^4 + y^3/x^6);
I think it reduces to this:
(after dividing by y/x,) How to find out if there is a constant 'a' for
which
a substitution H=y/x^a leads to an expression free of x and y?

Best,
NB

*************************************
Theorem.
Expresion u(x,y)  has the form  u(x,y)=h(y/x^a) * y/x  if only if
u(x,y) is solution of pde

a*y*diff(u(x,y),y)+x*diff(u(x,y),x)=(a-1)*u(x,y)      (1)

Proof.  General solution of equation (1) is
u(x,y)=F(y/x^a)*x^(a-1)=h(y/x^a)*(y/x^a)*x^(a-1)=h(y/x^a)*y/x.
Equation (1) can solve with Maple pdsolve.

Examples:
(%i1) pde:a*y*diff(u(x,y),y)+x*diff(u(x,y),x)=(a-1)*u(x,y)$

 1. Let u(x,y)=y^3/x^(3/2)=(y/x^(1/4))^2*y/x. Then a=1/4.

(%i2) subst([u(x,y)=y^3/x^(3/2),a=1/4],pde)$
(%i3) ev(%, nouns);
(%o3) -(3*y^3)/(4*x^(3/2))=-(3*y^3)/(4*x^(3/2))
(%i4) is(%);
(%o4) true

 2.  Let u(x,y)=(y/x)*sin(y^2/x^4 + y^3/x^6). Then a=2
(%i5) subst([u(x,y)=(y/x)*sin(y^2/x^4 + y^3/x^6),a=2],pde)$
(%i6) ev(%, nouns)$
(%i7) expand(%)$
(%i8) is(%);
(%o8) true

best

Aleksas D