1. **Problem:** Solve the initial value problem $y' = 10 - x$, with $y(0) = -1$. 2. **Formula and Explanation:** This is a first-order ordinary differential equation (ODE). The general solution can be found by integrating the right-hand side with respect to $x$: $$y = \int (10 - x) \, dx + C$$ where $C$ is the constant of integration. 3. **Intermediate Work:** $$y = 10x - \frac{x^2}{2} + C$$ Apply the initial condition $y(0) = -1$: $$-1 = 10 \cdot 0 - \frac{0^2}{2} + C \implies C = -1$$ 4. **Final Solution:** $$y = 10x - \frac{x^2}{2} - 1$$ This function satisfies the differential equation and the initial condition.