Solve the differential equation by separation of variables: $\frac{dy}{dx}=\frac{2y+5}{{8x+9}^2}$. Which of the following is the general solution?

• A. $y=-\frac{5}{2}+C{e}^{-\frac{2}{8x+9}}$
• B. $y=-\frac{5}{2}+C{e}^{\frac{2}{8x+9}}$
• C. $y=-\frac{5}{2}+C{8x+9}^2$
• D. $y=C(2y+5)\left({8x+9}^2\right)$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the differential equation is separable. Rewrite the equation as (dy/(2y + 5)) = (dx/(8x + 9)^2). This separates the variables y and x.
2. Step 2: Integrate both sides of the equation. For the left-hand side, integrate ∫(1/(2y + 5)) dy, and for the right-hand side, integrate ∫(1/(8x + 9)^2) dx.
3. Step 3: Solve the left-hand integral ∫(1/(2y + 5)) dy. Use the substitution u = 2y + 5, which simplifies the integral to ∫(1/u) du. The result is ln|u| = ln|2y + 5|.
4. Step 4: Solve the right-hand integral ∫(1/(8x + 9)^2) dx. Use the substitution v = 8x + 9, which simplifies the integral to ∫(1/v^2) dv. The result is -1/v = -1/(8x + 9).
5. Step 5: Combine the results of both integrals. You now have ln|2y + 5| = -1/(8x + 9) + C, where C is the constant of integration. Solve for y to express the general solution in terms of x.