Question

The general solution of the ordinary differential equation
\[ \frac{dy}{dx} = \log \left( x \frac{dy}{dx} - y \right) \] is ...........

• A. \( y = cx + e^c \), where \( c \) is an arbitrary constant
• B. \( y = cx^2 + e^c \), where \( c \) is an arbitrary constant
• C. \( y = cx - e^c \), where \( c \) is an arbitrary constant
• D. \( y = cx^2 - e^c \), where \( c \) is an arbitrary constant

Hint:

When solving differential equations involving logarithmic terms, substitution can often help simplify the equation, and subsequent differentiation can be used to obtain the general solution.

Answer 1

The Correct Option is C

Given the differential equation: \[ \frac{dy}{dx} = \log \left( x \frac{dy}{dx} - y \right) \] We can first make a substitution to simplify the equation. Let: \[ z = \frac{dy}{dx} \] Thus, the equation becomes: \[ z = \log \left( xz - y \right) \] Next, differentiate both sides with respect to \( x \) to eliminate the logarithmic term and proceed with solving for \( y \). After solving, we obtain the general solution as: \[ y = cx - e^c \] where \( c \) is an arbitrary constant. This is the correct solution to the given differential equation.