A simple differential equation question

[StephenD420]

Hello everyone

I having a little trouble with this simple problem:
Show that x*dF(x,y)/dy = y*dF(x,y)/dx is F(x,y) = A*exp(b/2*(x^2+y^2)) where A and b are constants using separation of variables.

x*df/dy = y*df/dx
x= y* df/dx*dy/df
x = y* dy/dx
x*dx = y*dy
integrate both sides x^2/2 = y^2/2 + c

so what do I do now to find the answer??

Thanks for any help you guys can give.
Stephen

 

[cragar]

so separation of variables allows you to guess that the solution is the product of a function of x and y. so try this and plug it in.

 

[StephenD420]

sorry..I am not sure what you mean. Could you explain it or show me the first couple of steps as I do not understand what you mean that the answer is a product of x and y...I know that that is why I separated the x's and y's so x's were on one side and the y's were on the other side of the equals sign.
Please explain what you mean by that...

 

[cragar]

like you would make the guess that F(x,y)=X(x)Y(y)
then find df/dy and df/dx and then plug this into your original equation.
so for example df/dx=dX/dx(Y(y))

 

[StephenD420]

ok if F=h(x)g(y)
then dF/dx = g(y)dh/dx
and dF/dy = h(x)dg/dy
so
x*dF/dy = y*dF/dx becomes
x*h(x)*dg/dy = y*g(y)dh/dx
putting all x' on one side and y's on the other gives
x*h(x)*dx/dh = y*g(y)*dy/dg

now what?

 

[cragar]

now subtract the y*g(y)*dy/dg to the other side. and then divide everything by
h(x) and g(y) now set each of the expressions being added together equal to a constant and then solve these 2 ode's.

 

[StephenD420]

ok so
x*h(x)*dx/dh = y*g(y)*dy/dg
becomes when you subtract and divide by h(x) and g(x) gives
x*h(x)*dx/dh - y*g(y)*dy/dg = 0 becomes
x/g(x)*dx/dh -y/h(x) *dy/dg = 0

so now what?? I am confused as to what two equations you mean and how to go from here.
Do you mean
x*dx/dh=c1
y*dy/dg=c2
then
x^2/2 =c1*h(x) + C3
y^2/2 = c2*g(x) + C4
now what??
Thanks.
Stephen

 

[cragar]

x/g(x)*dx/dh -y/h(x) *dy/dg = 0
x/g(x)*dx/dh=C and y/h(x) *dy/dg=D
so now you have x(dx/dh)=c(g(x)) and y(dy/dg)=D(h(y))
so now you have a derivative of a function equal to another number times itself.
this is usually an exponential function. I think i got the g(y) and h(x)'s mixed up
also maybe Google some examples of separation of variables