A student asserts that the function y=ln∣x∣ is a solution to the differential equation dxdy=x1 on the domain x=0. Which of the following best explains why the assertion is incomplete?
(a) Because ln∣x∣ satisfies dxdy=x1 only for x>0; for x<0 its derivative is −x1.
(b) A multiplicative constant should appear, so the general solution is y=Cln∣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∣ is x21, not x1.