1. **Stating the problem:** We are given the differential equation $$y' = k + e^{\frac{a}{r}}$$ and asked to find the function $$y$$ by integrating with respect to $$w$$. 2. **Formula used:** To find $$y$$ from $$y'$$, we use the integral: $$y = \int y' \, dw$$ 3. **Integration steps:** Given: $$y' = k + e^{\frac{a}{r}}$$ Integrate both terms separately: $$y = \int k \, dw + \int e^{\frac{a}{r}} \, dw$$ Since $$k$$ and $$e^{\frac{a}{r}}$$ are constants with respect to $$w$$ (no $$w$$ in the exponents or coefficients), the integrals are: $$\int k \, dw = k w$$ $$\int e^{\frac{a}{r}} \, dw = e^{\frac{a}{r}} w$$ 4. **Final solution:** $$y = k w + e^{\frac{a}{r}} w + C = w \left(k + e^{\frac{a}{r}}\right) + C$$ where $$C$$ is the constant of integration. **Note:** The expression $$\frac{e^{\frac{a}{r}}}{\frac{1}{r}}$$ mentioned is incorrect because $$e^{\frac{a}{r}}$$ is constant with respect to $$w$$, so the integral is simply $$e^{\frac{a}{r}} w$$. This completes the solution.