The differential equation dy dx = y x

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

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. (1) Identify all values of

y

for which the right-hand side of the differential equation is defined. Justify your conclusion. (2) 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. (3) 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.

Related theory

Introduction to Differential Equations
Foundations
Separation of Variables
Solutions
Variable separable equations
Differential equations with variables separable
Differential Equations

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