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

That is a different problem.  solution(h(x*y)*y/x,x,y) has no solution in the form required by this solution() function, at least I don?t know of any. On the other hand h(x^a*y)*y/x has a solution at [a = ?a, f(x) = h(x)]. I think you need a different solution() function.  BTW solution() is not a great name for this function.

Rich



From: Aleksas Domarkas 
Sent: Friday, December 14, 2012 1:28 AM
To: maxima 
Subject: finding out if expr has the form F(y/x^a)

Proposition.
Expresion u(x,y)  has the form  u(x,y)=h(x * y) * y/x  is equivalent to:
u(x,y) is solution of pde  

y*diff(u(x,y),y)-x*diff(u(x,y),x)=2*u(x,y)      (1)

Proof.  General solution of equation (1) is u(x,y)=F(x*y)/x^2=h(x*y)*(x*y)/x^2=h(x*y)*y/x. 
Equation (1) can solve with Maple pdsolve.

For example, we test this for u(x,y)=sin(x*y)*y/x

(%i1) pde:y*diff(u(x,y),y)-x*diff(u(x,y),x)=2*u(x,y);
(%o1) ('diff(u(x,y),y,1))*y-x*('diff(u(x,y),x,1))=2*u(x,y)
(%i2) subst(u(x,y)=sin(x*y)*y/x,pde);
(%o2) y*('diff((y*sin(x*y))/x,y,1))-x*('diff((y*sin(x*y))/x,x,1))=(2*y*sin(x*y))/x
(%i3) ev(%, nouns);
(%o3) y*(sin(x*y)/x+y*cos(x*y))-x*((y^2*cos(x*y))/x-(y*sin(x*y))/x^2)=(2*y*sin(x*y))/x
(%i4) ratsimp(%);
(%o4) (2*y*sin(x*y))/x=(2*y*sin(x*y))/x
(%i5) is(%);
(%o5) true

best 

Aleksas D



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