1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \frac{1}{\sqrt{x+2}}$$ with the initial condition $$y(2) = -1$$. We need to find the function $$y(x)$$. 2. **Recall the formula:** To solve for $$y(x)$$, integrate both sides with respect to $$x$$: $$y = \int \frac{1}{\sqrt{x+2}} \, dx + C$$ 3. **Integration:** Use substitution to integrate: Let $$u = x + 2$$, so $$du = dx$$. The integral becomes: $$\int \frac{1}{\sqrt{u}} \, du = \int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} + C = 2\sqrt{u} + C$$ 4. **Back-substitute:** Replace $$u$$ with $$x+2$$: $$y = 2\sqrt{x+2} + C$$ 5. **Apply initial condition:** Use $$y(2) = -1$$ to find $$C$$: $$-1 = 2\sqrt{2+2} + C = 2\sqrt{4} + C = 2 \times 2 + C = 4 + C$$ Solve for $$C$$: $$C = -1 - 4 = -5$$ 6. **Final solution:** $$y = 2\sqrt{x+2} - 5$$