ecuació diferencial exponencial de potencial

d_{x}[y(x)] = (y/x)·( m+x^{m} )

y(x) = x^{m}·e^{(x^{m}/m)}

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

Sigui y(x) = u(x)·x ==> 

u(x)+x·d_{x}[u(x)] = mu(x)+x^{m}u(x)

x·d_{x}[u(x)] = ( m+x^{m}+(-1) )·u(x)

ln(u(x)) = (m+(-1))·ln(x)+(x^{m}/m)

y(x) = ( x^{m+(-1)}·e^{(x^{m}/m)} )·x

d_{x}[y(x)] = (y/x)·( 1+a·(y/x)^{m} )

Sigui y(x) = u(x)·x ==>

x·d_{x}[u(x)] = a·( u(x) )^{m+1}

(-1)·(1/m)·(1/a)·( 1/( u(x) )^{m} ) = ln(x)

y(x) = ( ( (-1)·(1/m)·(1/a) )^{(1/m)}·( ln(x) )^{(-1)·(1/m)} )·x

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

... ( (-1)·(1/m)·(1/a) )^{(1/m)}·( (-1)·(1/m)·( ln(x) )^{(-1)·(1/m)+(-1)}+( ln(x) )^{(-1)·(1/m)} )

d_{x}[y(x)] = (y/x)·( m+( m/ln( x^{m}/m ) ) )

y(x) = x^{m}·ln( x^{m}/m )

d_{x}[ ln[m]( x^{m}/m ) ] = ( mx^{m+(-1)}·ln( x^{m}/m )+mx^{m+(-1)} )

Sigui y(x) = u(x)·x ==>

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

ln(u(x)) = (m+(-1))·ln(x)+ln(ln( x^{m}/m ))

y(x) = x^{m+(-1)}·ln( x^{m}/m )·x