ecuació diferencial amb potencia integral teorema de transformació

d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = k
y(x) = x


d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = a
y(x) = ( a/k )^{(1/n)}x


d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = h(x)


y(x) = ∫ [ ( ∫ [ ( f(x) )^{(n+(-1))} ] d[x]·(n+(-1))·k )^{(-1)/(n+(-1))·f(x) ] d[x]


( ∫ [ ( f(x) )^{(n+(-1))} ] d[x]·(n+(-1))·k )^{(-1)/(n+(-1))}·d_{x}[f(x)] = h(x)


exponencial:
f(x) = ( e^{(n+(-1))k·x} )^{1/(n+(-1))}


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


potencial:
f(x) = ( (n+(-1))k·x )^{1/(n+(-1))}


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


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


y(x) = ∫ [ 2^{1/(n+(-1))}·( (n+(-1))k·x )^{(-1)/(n+(-1))} ] d[x]


2^{1/(n+(-1))}( (-1)+2^{(n+(-1))/(n+(-1))} ) = 2^{1/(n+(-1))}