Question:
The general solution of the differential equation
\[
(9x - 3y + 5) dy = (3x - y + 1) dx.
\]
The general solution of the differential equation
\[
(9x - 3y + 5) dy = (3x - y + 1) dx.
\]
Show Hint
Convert the differential equation into standard form before solving.
- Use the integrating factor method to simplify linear differential equations.
Updated On: Mar 21, 2025
- \( 4x - 3y - \log |12x - 4y + 7| = c \)
- \( 4x - 12y - \log |12x - 4y + 7| = c \)
- \( 4x - 12y + \log |6x - 2y + 7| = c \)
- \( 4x - 6y + \log |12x - 4y + 7| = c \)
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The Correct Option is B
Solution and Explanation
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 \( \boxed{4x - 12y - \log |12x - 4y + 7| = c} \).
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