Solve the given DE by using the substitution u = y'

(y + 1)y" = (y')^2

(y + 1)u' = u^2

u' = u*du/dy

(y + 1)u*du/dy = u^2

(y + 1)du/dy = u

1/(y + 1) dy = (1/u) du

ln(y + 1) = ln(u) + C

y + 1 = Cu

u = (y + 1)/C = dy/dx

dx = C/(y + 1) dy

x + C2 = C*ln(y + 1)

I'd appreciate it if someone could look over this and tell me if I did it right, thanks in advance.