Hola nuevamente

Dividimos por 8 x

y' - (1/(8x)) y = (y^-3) (-(x + 1)^(1/2)/(8x))

sustituimos por

v = y^(1 - (-3)) = y^4

Multiplicamos todo por 4 y^3

4 y^3 y' - (1/2) (1/x) y^4 = (-1/2) (x + 1)^(1/2) / x

Queda

v' - (1/2) (1/x) v = (-1/2) (x + 1)^(1/2) / x

Factor integrante

x^(-1/2)

Nos queda

x^(-1/2) dv/dx + (-1/2) x^(-3/2) v = (-1/2) (x + 1)^(1/2) x^(-3/2)

x^(-1/2) dv + (-1/2) x^(-3/2) dx v = (-1/2) (x + 1)^(1/2) x^(-3/2) dx

x^(-1/2) dv + d(x^(-1/2)) dv = (-1/2) (x + 1)^(1/2) x^(-3/2) dx

d (x^(-1/2) v) = (-1/2) (x + 1)^(1/2) x^(-3/2) dx

Solución

x^(-1/2) v = K - (1/2) ʃ (x + 1)^(1/2) x^(-3/2) dx

Solución en x;y

x^(-1/2) y^4 = K - (1/2) ʃ (x + 1)^(1/2) x^(-3/2) dx

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