Linear Differential B9Cf12

1. **State the problem:** We need to solve the differential equation $$\frac{dy}{dx} = -3y - \frac{2}{8x+5}$$. 2. **Rewrite the equation:** Write it in standard linear form: $$\frac{dy}{dx} + 3y = -\frac{2}{8x+5}$$ 3. **Identify integrating factor:** The integrating factor (IF) is given by: $$\mu(x) = e^{\int 3 \, dx} = e^{3x}$$ 4. **Multiply both sides by IF:** $$e^{3x} \frac{dy}{dx} + 3 e^{3x} y = -\frac{2 e^{3x}}{8x+5}$$ This simplifies to: $$\frac{d}{dx} \left( e^{3x} y \right) = -\frac{2 e^{3x}}{8x+5}$$ 5. **Integrate both sides:** $$e^{3x} y = \int -\frac{2 e^{3x}}{8x+5} \, dx + C$$ 6. **Integration:** The integral $$\int -\frac{2 e^{3x}}{8x+5} \, dx$$ is not elementary in simple closed form, so we leave it as an integral expression or use substitution methods if needed. 7. **Solve for y:** $$y = e^{-3x} \left( C - \int \frac{2 e^{3x}}{8x+5} \, dx \right)$$ **Final answer:** $$\boxed{y = e^{-3x} \left( C - \int \frac{2 e^{3x}}{8x+5} \, dx \right)}$$ This is the general solution expressed in terms of an integral.