2015-05-22 01:09:59 UTC
dy/dx = - sqrt(y - 1/x)
can a real solution y(x) be found for which:
lim(x->inf) y(x) = 0
Squaring both sides of the equation, I have found the "solution":
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 +
3386880/x^19 + 146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 + 1641030725533696/x^34 + 119809030723207168/x^37 +
which appears to be an asymptotic power series. From Wikipedia
"Typically, the best approximation is given when the series is
truncated at the smallest term." So as x->inf it does appear that y->0.
But I have not found a convergent power series solution nor any closed
form solution of the type described. Does one exist?