Eurler’s equation

Let k be any real number , For a given k and t, x, y > 0 we have the following equation :

F( t*x, t*y ) = t^k * F( x, y )

We can substitute t = 1/x

F ( x, y ) = x^k  * F(1 , y/x )

We can define g(z) = F( 1 , z)

then finally :

F ( x, y )  = x ^ k * g( y/x )

An example could be  : F( x, y ) = x^2 +  4  * y^2  + 3 x*y

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Basic functional equation

function

find at least on real function which is defined by  : f( k * x) = f (x) for all x Real and k real

  • Obviously all constants satisfies the equation above , but you can say more if f is continue at x=0 , then f is a constant function
  • a non trivial example of a solution of equation above could be : x → sin( 2π * ln(x) )           with k = e, you can notice that this function is not continue at x = 0