Mathematics

Problem

A student asserts that the function

y=\ln|x|

is a solution to the differential equation

\dfrac{dy}{dx}=\dfrac{1}{x}

on the domain

x\neq0

. Which of the following best explains why the assertion is incomplete? a) Because

\ln|x|

satisfies

\dfrac{dy}{dx}=\dfrac{1}{x}

only for

x>0

; for

x<0

its derivative is

-\dfrac{1}{x}

. b) A multiplicative constant should appear, so the general solution is

y=C\ln|x|

. c) A constant of integration must be included, so the most general solution is

y=\ln|x|+C

. d) The derivative of

\ln|x|

\dfrac{1}{x^{2}}

, not

\dfrac{1}{x}