Question

If \( 2x - y + c \log(|x - 2y - 4|) = k \) is the general solution of \[ \frac{dy}{dx} = \frac{2x - 4y - 5}{x - 2y + 2} \] then \( c = \):

• A. 4
• B. 2
• C. 3
• D. -4

Hint:

Look for substitutions that reduce the differential equation to a separable or linear form.

Answer 1

The Correct Option is C

To solve \[ \frac{dy}{dx} = \frac{2x - 4y - 5}{x - 2y + 2} \] Use substitution: Let \( u = x - 2y \Rightarrow \frac{du}{dx} = 1 - 2\frac{dy}{dx} \). Substitute back into the equation and simplify. The differential equation transforms into one involving \( u \), which is separable. Solving leads to \[ 2x - y + 3 \log(|x - 2y - 4|) = k \] So, \( c = 3 \).