ecuacions diferencials

d_{x}[y(x)] = ( xy/(x^{2}+y^{2}) )

u+x·d_{x}[u(x)] = ( u/(1+u^{2}) )

( (-1)·(1/u^{3})+(-1)·(1/u) )·d_{x}[u(x)] = (1/x)

(1/2)·(1/u^{2}) = ln(ux)

(1/2)·x^{2} = y^{2}·ln(y)

(1/2)·x^{2} = ln[2](y)

y = e-ln[2]( (1/2)·x^{2} )

d_{x}[y(x)] = ( xy/(x^{2}+(-1)·y^{2}) )

u+x·d_{x}[u(x)] = ( u/(1+(-1)·u^{2}) )

( (1/u^{3})+(-1)·(1/u) )·d_{x}[u(x)] = (1/x)

(-1)·(1/2)·(1/u^{2}) = ln(ux)

(-1)·(1/2)·x^{2} = y^{2}·ln(y)

(-1)·(1/2)·x^{2} = ln[2](y)

y = e-ln[2]( (-1)·(1/2)·x^{2} )

d_{x}[e-ln[2]( (-1)·(1/2)·x^{2} )] = ...

... ( (-1)·x·e-ln[2]( (-1)·(1/2)·x^{2} ) )/( (-1)·x^{2}+( e-ln[2]( (-1)·(1/2)·x^{2} ) )^{2} )

d_{y}[ln[n](y)] = ( n·ln[n](y)+y^{n} )·(1/y)

d_{x}[e-ln[n](x)] = ( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} )

d_{xx}^{2}[e-ln[n](x)] = ( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} )^{2}+...

... (-1)·( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} )^{2}·...

... ( n+n·( e-ln[n](x) )^{n+(-1)}·( ( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} ) ) )

e-ln[n](x) = ...

... 1+x+( (1+(-1)·n)+(-1)·n )·(x^{2}/2!)+...