PDEchange PDE->ODE

[maciej...@gmail.com]

Hi,
I have the PDE
(x^2+y^2-2*A)*f(x,y)-1/4*diff(f(x,y),x,x)-1/4*diff(f(x,y),y,y)=0
Using the substitution z=2*(x^2+y^2), f(x,y)->f(z)
I wolud like to get the following ODE
(1/4*z-A)*f(z)-z*diff(f(z),z,z)-diff(f(z),z)=0

(A=const)

Using the chain rule it is easy to show that
df/dx=df/dz*dz/dx=4x*df/dz
and further that
d/dx(4x*df/dz)=4*df/dz+16x^2*(d^2F/dx^2)
For y variable we get in a similar way
diff(f(x,y),y,y)=4*df/dz+16y^2*(d^2F/dy^2)
Inserting the last second derivative to the first equation we get what we want.

The question is
how can I get this result using Maple commands.

Thank you in advance
Mak

[Mate]

Actually you are looking for a special solution f(x,y) =
g(2*(x^2+y^2))

> (x^2+y^2-2*A)*f(x,y)-1/4*diff(f(x,y),x,x)-1/4*diff(f(x,y),y,y):
> simplify(subs(f(x,y)=g(2*(x^2+y^2)), %)):
> convert(%,D):
> simplify(subs(y^2=z/2-x^2,%)):
> convert(%,diff);

1/2*g(z)*z-2*g(z)*A-2*diff(g(z),z)-2*diff(diff(g(z),z),z)*z