ecuacions diferencials

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

y = u+x

1+d_{x}[u] = ( 1/(1+(-1)·e^{u}) )

d_{x}[u] = ( e^{u}/(1+(-1)·e^{u}) )

(e^{-u}+(-1))·d_{x}[u] = 1

(-1)·e^{-u}+(-u) = x

(-1)·e^{x+(-y)} = y

(-1)·e^{x} = y·e^{y}

(-1)·e^{x} = e[1](y)

y = ln-e[1]( (-1)·e^{x} )

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

d_{x}[ln-e[n](x)] = ( (ln-e[n](x))/( n+ln-e[n](x) ) )·(1/x)

(-1)·e^{x} = y·e^{y}

(-1)·e^{x}·(1/y) = e^{y}

d_{x}[ln-e[1]((-1)·e^{x})] = ...

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

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

... ( 1/( (-1)·e^{x}+e^{y} ) )·(-1)·e^{x}

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

y = u+x

1+d_{x}[u] = ( 1/(1+e^{u}) )

d_{x}[u] = ( ((-1)·e^{u})/(1+e^{u}) )

(e^{-u}+1)·d_{x}[u] = (-1)

(-1)·e^{-u}+u = (-x)

(-1)·e^{x+(-y)} = (-y)

e^{x} = y·e^{y}

e^{x} = e[1](y)

y = ln-e[1]( e^{x} )