Solve the differential equation by separation of variables: $\frac{dy}{dx}=\frac{2y+7}{8x+9}$. Which of the following is the general solution?

• A. $8y+28=2x+9+C$
• B. $8y+28=2\ln |8x+9|+C$
• C. $2y+7=\ln |8x+9|+C$
• D. $2y+7=8x+9+C$

Answer: (B)

Verified step by step guidance

1. Step 1: Start by identifying that the differential equation can be solved using the method of separation of variables. The equation is given as \( \frac{dy}{dx} = \frac{2y + 7}{8x + 9} \). Rewrite it to separate the variables \( y \) and \( x \).
2. Step 2: Rearrange the equation to isolate \( y \) terms on one side and \( x \) terms on the other. Multiply both sides by \( 8x + 9 \) and divide by \( 2y + 7 \), resulting in \( \frac{dy}{2y + 7} = \frac{dx}{8x + 9} \).
3. Step 3: Integrate both sides of the equation. For the left-hand side, integrate \( \int \frac{1}{2y + 7} \, dy \), and for the right-hand side, integrate \( \int \frac{1}{8x + 9} \, dx \). Use substitution if necessary to simplify the integrals.
4. Step 4: Solve the integrals. The left-hand side becomes \( \frac{1}{2} \ln|2y + 7| \), and the right-hand side becomes \( \frac{1}{8} \ln|8x + 9| \). Combine the results and include the constant of integration \( C \).
5. Step 5: Multiply through by constants to simplify the expression into the general solution form. After simplification, the general solution is \( 8y + 28 = 2 \ln|8x + 9| + C \).