Question

The general solution of the differential equation \[ \frac{dy}{dx} = \frac{2xy-4x+y-2}{2xy+x-4y-2} \] is:

• A. \( 5(y-x)+2\log\left|\frac{y-2}{x-2}\right|=c \)
• B. \( 2(y-x)-5\log\left|\frac{y-2}{x-2}\right|=c \)
• C. \( 2(y-x)+5\log\left|\frac{y-2}{x-2}\right|=c \)
• D. \( 5(y-x)-2\log\left|\frac{y-2}{x-2}\right|=c \)

Hint:

For solving first-order differential equations of this form, use appropriate substitutions like \( v = y - x \) to simplify the equation before integrating.

Answer 1

The Correct Option is C

Step 1: Identifying the equation type Rewriting the given equation: \[ \frac{dy}{dx} = \frac{2xy - 4x + y - 2}{2xy + x - 4y - 2} \] This is a first-order differential equation. Using the substitution \( v = y - x \), meaning \( \frac{dv}{dx} = \frac{dy}{dx} - 1 \), simplifies the equation. Step 2: Substituting \( v = y - x \) Rewriting in terms of \( v \): \[ \frac{dv}{dx} + 1 = \frac{2x(v + x) - 4x + (v + x) - 2}{2x(v + x) + x - 4(v + x) - 2} \] Step 3: Solving for \( v \) Using separation of variables and integrating both sides carefully, the general solution simplifies to: \[ 2(y - x) + 5\log\left|\frac{y - 2}{x - 2}\right| = c \] Thus, the correct answer is: \[ 2(y - x) + 5\log\left|\frac{y - 2}{x - 2}\right|=c \]