Question

show that y=lnx is a sol'n of xy''+y'=0 the interval(0,∞). but is not a sol'n on (-∞,∞)....pls answer

Answer 1

domain of x in lnx?

Answer 2

there's no domain that's the equation

Answer 3

(-inf, +inf) includes zero

log(-x) = log(x) + i (pi + 2n pi)
looking from here, the negative should be solution of it.

Answer 4

lnx is meaningful, only when x ranges from (0, +infinity).
differentiation is allowed only when a function exists.
lnx does not exists in(-infinity, 0).

Answer 5

just differentiate above, you will note that it satisfies the above DE.

Answer 6

it does not satisfies when x is -ve, because the function lnx does not exist in -ve real.

Answer 7

ln(-x) = complex valued (and also multivalued)
if you choose one particular branch, it's a function.