ecuacions diferencials

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

y = u+x

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

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

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

(-u)+(1/2)·ln(1+2e^{u}) = x

(1/2)·ln(1+2e^{y+(-x)}) = y

1+2e^{y+(-x)} = e^{2y}

e^{y} = (1/2)·( 2e^{(-x)}+( 4e^{(-2)x}+4 )^{(1/2)} )

e^{y} = ( e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )

y = ln( e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )

e^{y}+(-1)·e^{(-x)} = ( e^{(-2)x}+1 )^{(1/2)}

1+2e^{y+(-x)} = e^{2y}

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

... ( (-1)+(-1)·e^{(-x)}(e^{(-2)x}+1)^{(-1)(1/2)} )/( 1+e^{x}(e^{(-2)x}+1)^{(1/2)} ) = ...

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

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

... ( (-1)·e^{y}/(e^{x}e^{2y}+(-1)·e^{y}) ) ...

... ( (-1)·e^{y}/(e^{x}(1+2e^{y+(-x)})+(-1)·e^{y}) )

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

y = u+x

1+d_{x}[u] = ( e^{u}/(1+(-1)·e^{u}) )

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

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

(-u)+(1/2)·ln(1+(-2)e^{u}) = x

(1/2)·ln(1+(-2)e^{y+(-x)}) = y

1+(-2)e^{y+(-x)} = e^{2y}

e^{y} = (1/2)·( (-2)·e^{(-x)}+( 4e^{(-2)x}+4 )^{(1/2)} )

e^{y} = ( (-1)·e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )

y = ln( (-1)·e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )

e^{y}+e^{(-x)} = ( e^{(-2)x}+1 )^{(1/2)}

1+(-2)e^{y+(-x)} = e^{2y}

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

... ( 1+(-1)·e^{(-x)}(e^{(-2)x}+1)^{(-1)(1/2)} )/( (-1)+e^{x}(e^{(-2)x}+1)^{(1/2)} ) = ...

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

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

... ( e^{y}/(e^{x}e^{2y}+e^{y}) ) ...

... ( e^{y}/(e^{x}(1+(-2)e^{y+(-x)})+e^{y}) )