Mathematics

Problem

The differential equation

\frac{dy}{dx}=\frac{\ln y}{x}, \quad x>0,

is to be analyzed with respect to the domain of its solutions. a. Identify all values of

y

for which the right-hand side of the differential equation is defined. Justify your conclusion. b. Suppose separation of variables leads to the implicit family of relations

y\ln y-y=x\ln x-x+C,

where

C

is an arbitrary constant. Explain why any solution obtained from this relation must satisfy

y>0

for every

x

in its domain. c. For a fixed constant

C

, describe qualitatively (without solving explicitly for

y

) how the sign of

\dfrac{dy}{dx}

depends on the value of

y

. Use this information to discuss whether solution curves can cross the line

y=1

. Provide reasoning based on the differential equation rather than on an explicit solution.

Browse the problem library