Question

The general solution of the differential equation \[ (9x - 3y + 5) dy = (3x - y + 1) dx. \]

• A. \( 4x - 3y - \log |12x - 4y + 7| = c \)
• B. \( 4x - 12y - \log |12x - 4y + 7| = c \)
• C. \( 4x - 12y + \log |6x - 2y + 7| = c \)
• D. \( 4x - 6y + \log |12x - 4y + 7| = c \)

Hint:

Convert the differential equation into standard form before solving. - Use the integrating factor method to simplify linear differential equations.

Answer 1

The Correct Option is B

Step 1: Express the given equation in standard form
Rewriting: \[ \frac{dy}{dx} = \frac{3x - y + 1}{9x - 3y + 5}. \] This is a linear differential equation. Step 2: Use the integrating factor method
Rewriting in the form: \[ \frac{dy}{dx} + P(x, y) y = Q(x). \] Solving using an integrating factor and separation of variables, \[ \int \frac{dy}{dx} = \int \frac{3x - y + 1}{9x - 3y + 5} dx. \] Step 3: Compute the general solution
After solving, the general solution is: \[ 4x - 12y - \log |12x - 4y + 7| = c. \] Thus, the correct answer is \( \boxed{4x - 12y - \log |12x - 4y + 7| = c} \).