Question

The general solution of the differential equation \( 2dx + dy = (6xy + 4x - 3y)dx \) is

• A. \( 2\log|2x-1| = 3y^2 + 4y + c \)
• B. \( \log|3y+2| = 3x^2 - 3x + c \)
• C. \( \log|3y+2| = x^2 - x + c \)
• D. \( \log|2x-1| = 3y^2 - 4y + c \)

Hint:

When you see a mix of \( xy \), \( x \), and \( y \) terms like \( Axy + Bx + Cy + D \), try grouping terms to factorize it into \( (ax+b)(cy+d) \). This often makes the differential equation variable separable.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

To find the general solution of the given differential equation, we need to simplify it into a standard form. We will attempt to use the method of separation of variables, where we separate the terms involving \( x \) and \( y \) on opposite sides of the equation.
Step 2: Key Formula or Approach:

1. Rearrange the terms to express \( \frac{dy}{dx} \). 2. Factorize the expression to separate variables. 3. Integrate both sides using standard integration formulas:
- \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \)
- \( \int \frac{1}{ax+b} \, dx = \frac{1}{a} \log|ax+b| \)
Step 3: Detailed Explanation:

Given the differential equation: \[ 2dx + dy = (6xy + 4x - 3y)dx \] First, rearrange the terms to isolate \( dy \): \[ dy = (6xy + 4x - 3y)dx - 2dx \] \[ dy = (6xy + 4x - 3y - 2)dx \] Dividing by \( dx \): \[ \frac{dy}{dx} = 6xy + 4x - 3y - 2 \] Now, factorize the Right Hand Side (RHS) by grouping terms: \[ \text{RHS} = 2x(3y + 2) - 1(3y + 2) \] \[ \text{RHS} = (2x - 1)(3y + 2) \] So the equation becomes: \[ \frac{dy}{dx} = (2x - 1)(3y + 2) \] Separate the variables \( x \) and \( y \): \[ \frac{1}{3y + 2} \, dy = (2x - 1) \, dx \] Integrate both sides: \[ \int \frac{1}{3y + 2} \, dy = \int (2x - 1) \, dx \] Using the integration formulas: \[ \frac{1}{3} \log|3y + 2| = \frac{2x^2}{2} - x + C_1 \] \[ \frac{1}{3} \log|3y + 2| = x^2 - x + C_1 \] Multiply the entire equation by 3 to eliminate the fraction: \[ \log|3y + 2| = 3(x^2 - x) + 3C_1 \] Let \( c = 3C_1 \) be the arbitrary constant. \[ \log|3y + 2| = 3x^2 - 3x + c \]
Step 4: Final Answer:

The general solution is \( \log|3y+2| = 3x^2 - 3x + c \).