Subject: finding out if expr has the form F(y/x^a)
From: Aleksas Domarkas
Date: Fri, 14 Dec 2012 08:28:58 +0200
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